WISKUNDE
GRAAD 11
NOG OEFENINGE
  
Reduksie Formules.
  
MATHEMATICS
GRADE 11
MORE EXERCISES
  
Reduction Formulae.
  
  
                 Pas die volgende reëls toe as jy die
                 formules gebruik :
             1. Die teken van die funksie word bepaal deur
                 die oorspronklike funksie in die
                 oorspronklike kwadrant deur die CAST
                 reël te gebruik.
             2. Wanneer die waarde van die funksies van
                 (180° ± θ) of (360° − θ) bepaal word, verander
                 die funksie nooit nie maar die
                 teken mag verander.
             3. Wanneer die waarde van die funksies van
                 (90° ± θ) of (270° ± θ) bepaal word, verander
                 die funksie na sy ko-funksie en die
                 teken mag verander.
  
                 Apply the following rules when using the
                 reduction formulae :
             1. The sign of the function is determined
                 from the original function in the
                 original quadrant by using the CAST rule.
  
             2. When determining the value of the functions
                 of (180° ± θ) or (360° − θ),
                 the function never changes but
                 the sign may change.
             3. When determining the value of the functions
                 of (90° ± θ) or (270° ± θ),
                 the function changes to its co-function and
                 the sign may change.
  
Vraag / Question  1
  
             1.1    sin (180° − θ)  = sin θ       ┃ 180° − θ : naam onveranderd; hoek in kwadrant 2; sin (180° − θ) > 0   [CAST]
                                                               ┃ 180° − θ : name unchanged; angle in quadrant 2; sin (180° − θ) > 0   [CAST]
                                                                                                                                                                 [ 1.1 ]
  
             1.2  cos (180° − θ) = ─ cos θ        ┃ 180° − θ : name unchanged; angle in quadrant 2; cos (180° − θ) < 0 [CAST]
                                                                   ┃ 180° − θ : naam onveranderd; hoek in kwadrant 2; cos (180° − θ) < 0 [CAST]
                                                                                                                                                                 [ 1.2 ]
  
             1.3  tan (360° − θ) = ─ tan θ       ┃ 360° − θ : naam onveranderd; hoek in kwadrant 4; tan (360° − θ) < 0 [CAST]
                                                                 ┃ 360° − θ : name unchanged; angle in quadrant 4; tan (360° − θ) <  [CAST]
                                                                                                                                                                 [ 1.3 ]
  
             1.4  sec (360° + θ) = sec θ        ┃ 360° + θ : naam onveranderd; hoek in kwadrant 1; sec (360° + θ) > 0 [CAST]
                                                               ┃ 360° + θ : name unchanged; angle in quadrant 1; sec (360° + θ) > 0 [CAST]
                                                                                                                                                                [ 1.4 ]
  
             1.5  cosec (360° − θ) = − cosec θ
                                             ┃ 360° − θ : naam onveranderd; hoek in kwadrant 4; cosec (360° + θ) < 0 [CAST]
                                             ┃ 360° − θ : name unchanged; angle in quadrant 4; cosec (360° + θ) < 0 [CAST]
                                                                                                                                                                [ 1.5 ]
  
             1.6  cot (180° + θ) = cot θ       ┃ 180° + θ : naam onveranderd; hoek in kwadrant 3; cot (180° + θ) > 0 [CAST]
                                                             ┃ 180° + θ : name unchanged; angle in quadrant 3; cot (180° + θ) > 0 [CAST]
                                                                                                                                                                [ 1.6 ]
  
             1.7  tan (180° − θ)  =  ─ tan θ                     [ 1.7 ]
  
             1.8  sin (180° + θ) = − sin θ                        [ 1.8 ]
  
             1.9  cos (360° − θ) = cos θ                         [ 1.9 ]
  
  
           1.10  cosec (180° − θ) = cosec θ               [ 1.10 ]
  
  
           1.11  cot (360° − θ) = ─ cot θ                    [ 1.11 ]
  
  
           1.12  tan (180° + θ) = tan θ                        [ 1.12 ]
  
  
           1.13  sin (90° − θ) = cos θ           ┃ 90° − θ : naam verander; hoek in kwadrant 1; cot (90° − θ) > 0 [CAST]
                                                                  ┃ 90° − θ : name changes; angle in quadrant 1; sin (90° − θ) > 0 [CAST]                         [ 1.13 ]
  
  
           1.14  cos (90° + θ) = − sin θ        ┃ 90° + θ : naam verander; hoek in kwadrant 2; cos (90° + θ) < 0 [CAST]
                                                                  ┃ 90° + θ : name changes; angle in quadrant 2; cos (90° + θ) < 0 [CAST]                       [ 1.14 ]
  
  
           1.15  tan (270° − θ) = cot θ        ┃ 270° − θ : naam verander; hoek in kwadrant 3; tan (270° − θ) > 0 [CAST]
                                                                ┃ 270° − θ : name changes; angle in quadrant 3; tan (270° − θ) > 0 [CAST]                      [ 1.15 ]
  
  
           1.16  cot (270° + θ) = − tan θ        ┃ 270° + θ : naam verander; hoek in kwadrant 4; cot (270° + θ) < 0 [CAST]
                                                                   ┃ 270° + θ : name changes; angle in quadrant 4; cot (270° + θ) < 0 [CAST]                   [ 1.16 ]
  
  
           1.17  cosec (90° − θ) = sec θ        ┃ 90° − θ : naam verander; hoek in kwadrant 1; cosec (90° − θ) > 0 [CAST]
                                                                    ┃ 90° − θ : name changes; angle in quadrant 1; cosec (90° − θ) > 0 [CAST]                 [ 1.17 ]
  
  
           1.18  sec (90° + θ) = − cosec θ        ┃ 90° + θ : naam verander; hoek in kwadrant 2; sec (90° + θ) < 0 [CAST]
                                                                       ┃ 90° + θ : name changes; angle in quadrant 2; sec (90° + θ) < 0 [CAST]                  [ 1.18 ]
  
  
           1.19  tan (90° + θ) = − cot θ                    [ 1.19 ]
  
  
           1.20  sec (270° + θ) = cosec θ               [ 1.20 ]
  
  
           1.21  sin (270° − θ) = − cos θ                 [ 1.21 ]
  
  
           1.22  cos (270° + θ) = − sin θ                 [ 1.22 ]
  
  
           1.23  sin (90° + θ) = cos θ                      [ 1.23 ]
  
  
           1.24  cos (90° − θ) = sin θ                      [ 1.24 ]
  
  
Vraag / Question  2
  
  
           2.1  sin 70° = sin (90° − 20°)
                               = cos 20°                                [ 2.1 ]
  
  
           2.3  cosec 200° = cosec (180° + 20°)
                                       =  − cosec 20°
                                                                                [ 2.3 ]
  
  
           2.5  cot 250° = cot (270° − 20°)
                                  =  tan 20°                             [ 2.5 ]
  
  
           2.7  sin 200° = sin (180° + 20°)
                                 =  − sin 20°                           [ 2.7 ]
  
  
           2.9  tan 290° = tan (270° + 20°)
                                  =  − cot 20°                         [ 2.9 ]
  
  
         2.11  cot 110° = cot (90° + 20°)
                                  =  − tan 20°                      [ 2.11 ]
  
  
         2.13  cot 200° = cot (180° + 20°)
                                  =  cot 20°                          [ 2.13 ]
  
  
         2.15  sin 110° = c (90° + 20°)
                                  =  cos 20°                         [ 2.15 ]
  
  
         2.17  tan 110° = tan (90° + 20°)
                                  =  − cot 20°                       [ 2.17 ]
  
  
  
           2.2  cos 110° = cos (90° + 20°)
                                   =  − sin 20°                      [ 2.2 ]
  
  
           2.4  sec 340° = sec (360° − 20°)
                                   =  sec 20°                        [ 2.4 ]
  
  
  
           2.6  tan 380° = tan (360° + 20°)
                                  =  tan 20°                          [ 2.6 ]
  
  
           2.8  cos 160° = cos (180° − 20°)
                                   =  − cos 20°                    [ 2.8 ]
  
  
         2.10  cosec 70° = cosec (90° − 20°)
                                     =  sec 20°                   [ 2.10 ]
  
  
         2.12  sec 250° = sec (270° − 20°)
                                   =  − cosec 20°              [ 2.12 ]
  
  
         2.14  cosec 250° = cosec (270°−20°)
                                       =  − sec 20°               [ 2.14 ]
  
  
         2.16  cos 70° = cos (90° − 20°)
                                 =  sin 20°                         [ 2.16 ]
  
  
         2.18  sec 70° = sec (90° − 20°)
                                 =  cosec 20°                    [ 2.18 ]
  
  
Vraag / Question  3
  
  
           3.1  cos 50° = cos (90° − 40°)
                                =  sin 40°                                  [ 3.1 ]
  
  
           3.3  cot 310° = cot (270° + 40°)
                                  =  − tan 40°                             [ 3.3 ]
  
  
           3.5  sin 230° = sin (270° − 40°)
                                 =  − cos 40°                            [ 3.5 ]
  
  
           3.7  cos 310° = cos (270° + 40°)
                                   =  sin 40°                                [ 3.7 ]
  
  
           3.9  cosec 130° = cosec (90°−40°)
                                       =  sec 40°                           [ 3.9 ]
  
  
         3.11  cos 130° = cos (90° + 40°)
                                   =  − sin 40°                           [ 3.11 ]
  
  
         3.13  cot 50° = cot (90° − 40°)
                                =  tan 40°                                 [ 3.13 ]
  
  
         3.15  sec 140° = sec (180° − 40°)
                                   =  − sec 40°                         [ 3.15 ]
  
  
         3.17  sin 320° = sin (360° − 40°)
                                 =  − sin 40°                            [ 3.17 ]
  
  
  
           3.2  cosec 320° = cosec (360°−40°)
                                       =  − cosec 40°              [ 3.2 ]
  
  
           3.4  tan 130° = tan (90° + 40°)
                                  =   − cot 40°                       [ 3.4 ]
  
  
           3.6  sec 400° = sec (360° + 40°)
                                   =   sec 40°                        [ 3.6 ]
  
  
           3.8  sec 230° = sec (270° − 40°)
                                   =  − cosec 40°                  [ 3.8 ]
  
  
         3.10  sin 220° = sin (180° + 40°)
                                  =   − sin 40°                    [ 3.10 ]
  
  
         3.12  tan 230° = tan (270° − 40°)
                                  =   cot 40°                       [ 3.12 ]
  
  
         3.14  cosec 310° = cosec (270°+40°)
                                       =   − sec 40°              [ 3.14 ]
  
  
         3.16  cos 230° = cos (270° − 40°)
                                   =  − sin 40°                     [ 3.16 ]
  
  
         3.18  tan 140° = tan (180° − 40°)
                                  =   − tan 40°                     [ 3.18 ]
  
  
Vraag / Question  4
  
  
           4.1  sin 155° = sin (180° − 25°)
                                 =  sin 25°
  
                             OF  /  OR
  
                  sin 155° = sin (90° + 65°)
                                 =  cos 65°                              [ 4.1 ]
  
  
           4.3  tan 145° = tan (180° − 45°)
                                 =   − tan 45°
  
                             OF  /  OR
  
                  tan 145° = tan (90° + 55°)
                                 =  − cot 55°                           [ 4.3 ]
  
  
           4.5  sec 298° = sec (360° − 62°)
                                   =   sec 62°
  
                             OF  /  OR
  
                  sec 298° = sec (270° + 28°)
                                  =  cosec 28°                         [ 4.5 ]
  
  
           4.7  sin 138° = sin (180° − 42°)
                                 =   sin 42°
  
                             OF  /  OR
  
                  sin 138° = sin (90° + 48°)
                                 =  cos 48°                              [ 4.7 ]
  
  
           4.9  cot 118° = cot (180° − 62°)
                                  =   − cot 62°                         [ 4.9 ]
  
  
         4.11  cot 288° = cot (360° − 72°)
                                  =   − cot 72°                       [ 4.11 ]
  
  
         4.13  cos 308° = cos (360°−52°)
                                   =   cos 52°                        [ 4.13 ]
  
  
         4.15  sec 218° = sec (180° + 38°)
                                   =   − sec 38°                     [ 4.15 ]
  
  
         4.17  cosec 232° = cosec (180°+52°)
                                       =   − cosec 52°             [ 4.17 ]
  
  
  
           4.2  cosec 222° = cosec (180°+42°)
                                       =   − cosec 42°
  
                             OF  /  OR
  
                  cosec 222° = cosec (270°−48°)
                                      =  − sec 48°                  [ 4.2 ]
  
  
           4.4  cot 213° = cot (180° + 33°)
                                  =   cot 33°
  
                             OF  /  OR
  
                  cot 213° = cot (270° − 57°)
                                 =  tan 57°                           [ 4.4 ]
  
  
           4.6  cosec 124° = cosec (180°−56°)
                                       =   cosec 56°
  
                             OF  /  OR
  
                  cosec 124° = cosec (90°+34°)
                                      =  sec 34°                     [ 4.6 ]
  
  
           4.8  sec 318° = sec (360° − 42°)
                                   =   sec 42°
  
                             OF  /  OR
  
                  sec 318° = sec (270° + 48°)
                                   =  cosec 48°                     [ 4.8 ]
  
  
         4.10  cosec 346° = cosec (360°−14°)
                                       =   − cosec 14°          [ 4.10 ]
  
  
         4.12  tan 113° = tan (180° − 67°)
                                  =   − tan 67°                    [ 4.12 ]
  
  
         4.14  sin 150° = sin (180°−30°)
                                  =   sin 30°                        [ 4.14 ]
  
  
         4.16  cot 128° = cot (180° − 52°)
                                  =   − cot 52°                    [ 4.16 ]
  
  
         4.18  cos 142° = cos (180° − 38°)
                                   =   − cos 38°                 [ 4.18 ]
  
  
Vraag / Question  5
  
  
           Gebruik (180° ± 23°)  OF  (360° ± 23°) om die gegewe sin verhouding om te skakel na sin 23°
           Gebruik (90° ± 23°)  OF  (270° ± 23°) om die gegewe cos verhouding om te skakel na sin 23°
  
           Use (180° ± 23°)  OR  (360° ± 23°) to change the given sin ratio to sin 23°
           Use (90° ± 23°)  OR  (270° ± 23°) to change the given cos ration to sin 23°
  
  
           5.1  cos 67° = cos (90° − 23°)
                                =   sin 23°
                                =   p                                          [ 5.1 ]
  
  
           5.3  cos 293° = cos (270° + 23°)
                                  =  sin 23
                                  =  p                                           [ 5.3 ]
  
  
           5.5  cos 113° = cos (90° + 23°)
                                  =  − sin 23
                                  =  − p                                      [ 5.5 ]
  
  
           5.7  sin2 23° + cos2 23° = 1
                                      __________
                  cos 23° = √1 − sin2 23°
                                        ______
                                =  √1 − p2                                [ 5.7 ]
  
  
  
           5.2  sin 157° = sin (180° − 23°)
                                 =  sin 23
                                 =  p                              [ 5.2 ]
  
  
           5.4  sin 203° = sin (180° + 23°)
                                 =  − sin 23°
                                 =  − p                            [ 5.4 ]
  
  
           5.6  sin 337° = sin (360° − 23°)
                                 =  − sin 23
                                 =  − p                              [ 5.6 ]
  
  
  
           5.8  cos 247° = cos (270° − 23°)
− sin 23
                                 =  − p                              [ 5.8 ]
  
  
  
  
                                              1
           5.9  cosec 23° = ————
                                          sin 23°
  
                                           1
                                     =  ——
                                           p                              [ 5.9 ]
  
  
  
                                    sin 23°
          5.10  tan 23° = ————
                                    cos 23°
  
                                            p
                                 =  —————
                                        ______
                                      √1 − p2
  
                                             ______
                                        p √1 − p2
                                 =  ——————
                                          1 − p2
                                                               [ 5.10 ]
  
  
Antwoorde / Answers  6
  
  
           6.1  sec (180° + x) . cosec (180° ─ x) . cos (360° ─ x) . sin (180° + x)
  
                   =  ( ─sec x) . (cosec x) . (cos x) . ( ─ sin x)
  
  
                              1                  1
                   =   ─────  X  ─────  X  cos x  X  sin x
                           cos x           sin x
  
                   =  1                                                                                                                                [ 6. ]
  
  
  
           6.2  tan (180° + x) . cos (360° ─ x) . sec (90° + x)
  
  
                   =  ( tan x) . (cos x) . ( ─ cosec x)
  
  
                               sin x                            1
                   =  ─  ─────  X  cos x  X  ─────
                              cos x                          sin x
  
                   =  ─ 1                                                                                                                            [ 6. ]
  
  
  
           6.3  tan (180° + x)  tan (180° ─ x) + tan (360° + x)  tan (360° ─ x)  
  
                  =  ( tan x)  (─ tan x) + ( tan x) ─ (─ tan x)
  
  
                   =  4 tan x                                                                                                                       [ 6. ]
  
  
  
           6.4  sin (180° ─ x) . cos (90° ─ x)  cos (180° + x) . cos (360° ─ x)
  
                  =  ( sin x) . (sin x)  ( ─ cos x) . (cos x)
  
                  =  sin2 x + cos2 x
  
                   =  1                                                                                                                                [ 6. ]
  
  
  
           6.5  sin (90° + x) . cos (90° ─ x) + cos (90° + x) . sin (90° ─ x)
  
                  =  ( cos x) . (sin x) + ( ─ sin x) . (cos x)
  
                   =  0                                                                                                                                [ 6. ]
  
  
  
                        sin (360° ─ x) . tan (180° + x)
             6.6   ─────────────────────
                        cos x . cot (90° ─ x) . sec (180° + x)
  
  
                             ( ─ sin x) . (tan x)
                      =   ─────────────────────
                                                                  1
                             cos x . (tan x) .  ( ─  ──── )
                                                              cos x
  
  
                      =  sin x                                                                                                                       [ 6. ]
  
  
  
                        cos (180° + θ) . sin (180° ─ θ)
             6.7   ─────────────────────
                        cos (360° ─ θ) . tan (360° ─ θ)
  
  
                             ( ─ cos θ) . (sin θ)
                      =   ──────────────
                             (cos θ) . ( ─ tan θ)
  
  
                              sin θ           cos θ
                      =   ─────  X  ─────
                                1               sin θ
  
  
                      =  cos θ                                                                                                                      [ 6. ]
  
  
  
                        sin (360° + x) ÷ cos (90° ─ x) ÷ sin (180° ─ x)
             6.8   ───────────────────────────────
                        tan (180° + x) ÷ tan (360° + x) ÷ cos (270° ─ x)
  
  
                               (sin x) ÷ (sin x) ÷ (sin x)
                      =   ──────────────────
                             (tan x) ÷ (tan x) ÷ ( ─ sin x)
  
  
                                 1 ÷ (sin x)
                      =  ─  ─────────
                                 1 ÷ (sin x)
  
  
                      =  ─ 1                                                                                                                         [ 6. ]
  
  
  
                                  cos2 (180° ─ x)
             6.9   ─────────────────────
                        sin (360° ─ x) . sin (90° ─ x)
  
  
                                   ( ─ cos x)2
                      =   ─────────────
                             ( ─ sin x) . (cos x)
  
  
                                 cos x
                      =  ─  ─────
                                 sin x
  
  
                      =  ─ cot x                                                                                                                    [ 6. ]
  
  
  
                           cos (90° + α) . sin ( ─ α)
           6.10   ─────────────────────
                        sin (180° + α) . tan (360° ─ α)
  
  
                             ( ─ sin α) . ( ─ sin α)
                      =   ──────────────
                             ( ─ sin α) . ( ─ tan α)
  
  
                              sin α           cos α
                      =   ─────  X  ─────
                                1               sin α
  
  
                      =  cos α                                                                                                                      [ 6. ]
  
  
  
                           cos ( ─ β) . sin (180° + β)
           6.11   ─────────────────────
                        tan (β ─ 180°) . sin (β + 90°)
  
  
                                  (cos β) . ( ─ sin β)
                      =   ───────────────────
                              ( ─ tan (180° ─ β)) . (cos β)
  
  
                                  cos β . sin β
                      =   ───────────────────
                              ( ─ tan β) . cos β
  
  
                                  sin β           cos β
                      =  ─  ─────  X  ─────
                                    1               sin β
  
  
                      =  ─ cos β                                                                                                                   [ 6. ]
  
  
  
                               sin ( ─ α) . sec ( ─ α) . tan (180° ─ α)
           6.12   ───────────────────────────────
                        sec (360° ─ α) . cos  (270° ─ α) . tan (180° + α)
  
  
                              ( ─ sin α) . (sec α) . ( ─ tan α)
                      =   ─────────────────────
                                (sec α) . ( ─ sin α) . (tan α)
  
  
                      =  ─ 1                                                                                                                         [ 6. ]
  
  
Antwoorde / Answers  7
  
  
           7.1  sin 320° . tan 140° . cot 220°. sec 130°
  
  
                      =  sin (360° ─ 40°)   . tan (180° ─ 40°) . cot (180° + 40°) . sec (90° ─ 40°)
  
                      =  ( ─ sin 40°) . ( ─ tan 40°) . (cot 40°) . ( ─ cosec 40°)
  
                                                 sin 40°         cos 40°             1
                      =  ─ sin 40°  X  ─────  X  ──────  X  ─────
                                                cos 40°         sin 40°          sin 40°
  
  
                      =  ─ 1                                                                                                                                  [ 7. ]
  
  
  
             7.2  sin 200° . sec 110°  +  cos 160° . cosec 110°;
  
  
                      =  sin (180° + 20°)   . sec (90° + 20°)  +  cos (180° ─ 20°) . cosec (90° + 20°)
  
                      =  ( ─ sin 20°)   . ( ─ cosec 20°)  +  ( ─ cos 20°) . (sec 20°)
  
                      =  1  +  ( ─1)
  
                      =  0                                                                                                                                      [ 7. ]
  
  
  
             7.3  cos 290° . tan 200°  ─  cos 160°
  
  
                      =  cos (270° + 20°)   . tan (180° + 20°)  ─  cos (180° ─ 20°)
  
                      =  ( ─ sin 20°)   . ( ─ (tan 20°)  ─  ( ─ cos 20°)
  
                                              sin 20°
                      =  sin 20°  X  ─────  +  cos 20°
                                             cos 20°
  
  
                             sin2 20°        cos2 20°
                      =  ──────  +  ──────
                            cos 20°          cos 20°
  
  
                                1
                      =  ──────
                             cos 20°
  
  
                      =  sec 20°                                                                                                                           [ 7. ]
  
  
  
             7.4  cos 63° . cosec 207° . cot 27°
  
                      =  cos (90° ─ 27°) . cosec (180° + 27°)  .  cot 27°
  
                      =  (sin 27°)   . ( ─ cosec 27°) . (cot 27°)
  
                      =  ─ cot 27°                                                                                                                        [ 7. ]
  
  
  
             7.5  sin 125° . cot 145° . cosec 55° . sin 215°
  
                      =  sin (90° + 35°) . cot (180° ─ 35°) . cosec (90° ─ 35°) . sin (180° + 35°)
  
                      =  cos 35° . ( ─ cot 35°) . sec 35° . ( ─ sin 35°)
  
                                               cos 35°             1
                      =  cos 35°  X  ──────  X  ──────  X  sin 35°
                                               sin 35°          cos 35°
  
  
                      =  cos 35°                                                                                                                           [ 7. ]
  
  
  
                       cosec 110°
             7.6  ─────────
                       2 sec 20°
  
  
                            cosec (90° + 20°)
                      =  ────────────
                                2 sec 20°
  
  
                              sec 20°
                      =  ────────
                            2 sec 20°
  
  
                            1
                      =  ──
                            2                                                                                                                                   [ 7. ]
  
  
  
                       tan 340° . sin 72°                  tan (270° + 70°) . sin 72°
             7.7  ────────────      =      ─────────────────
                       cos 18° . cot 70°                  cos (90° ─ 72°) . cot 70°
  
  
                                                                    ( ─ cot 70°) . sin 72°
                                                         =      ─────────────────
                                                                    (sin 72°) . cot 70°
  
  
  
                                                         =  ─ 1                                                                                               [ 7. ]
  
  
  
                       sec 185° . sin 78°                        sec (180° + 5°)  . sin 78°
             7.8  ────────────      =      ───────────────────────
                       cos 348° . sec 175°               cos (270° + 78°) . sec (180° ─ 5°)
  
  
                                                                    ( ─ sec 5°) . sin 78°
                                                         =      ────────────────
                                                                    (sin 78°) . ( ─ sec 5°)
  
  
  
                                                         =  1                                                                                                  [ 7. ]
  
  
  
                       3 tan 65° . cosec 160°                         3 tan 65° . cosec (180° ─ 20°)
             7.9  ────────────────      =      ─────────────────────────
                       4 cosec 200° . cot 25°                    4 cosec (180° + 20°) . cot (90° ─ 65°)
  
  
                                                                              3 (tan 65°) . (cosec 20°)
                                                                   =      ────────────────
                                                                              4 ( ─ cosec 20°) . (tan 65°)
  
  
  
                                                                              3
                                                                   =  ─  ──
                                                                              4                                                                                 [ 7. ]
  
  
  
                                          cot 27° . tan 153°
           7.10    ────────────────────────────
                       sin 27° . cos 117°    cosec 207° . sin 333°
  
  
                                                              cot 27° . tan (180° ─ 27°)
                          =  ────────────────────────────────────────────
                                    sin 27° . cos (90° + 27°)  ─  cosec (180° + 27°) . sin (360° ─ 27°)
  
  
                                                              cot 27° . ( ─ tan 27°)
                          =  ─────────────────────────────────────
                                    sin 27° . ( ─ sin 27°)  ─  ( ─ cosec 27°) . ( ─ sin 27°)
  
  
                                          ─ 1
                          =  ─────────────
                                 ─ sin2 27°  ─  1
  
  
                                           1
                          =  ─────────────
                                 sin2 27° + 1                                                                                                              [ 7. ]
  
  
  
                            1
           7.11    ────── X cosec 128°  +  sec 232° . cos2 142°
                        cot 52°
  
  
                      =  tan 52° . cosec (180° ─ 52°)  +  sec (180° ─ 52°) . (cos (90° + 52°))2
  
                      =  tan 52° . cosec 52°  +  ( ─ sec 52°) . ( ─ sin 52°)2
  
                             sin 52°                1                   ( ─ 1 )            sin2 52°
                      =  ──────   X   ──────   +  ──────   X   ──────
                            cos 52°            sin 52°            cos 52°                  1
  
  
                                1                 sin2 52°
                      =  ──────   ─  ──────
                            cos 52°           cos 52°
  
  
                             1  ─  sin2 52°
                      =  ──────────
                                 cos 52°
  
  
                      =  cos 52°                                                                                                                           [ 7. ]
  
  
  
                               sin 108° . tan 72° . sin 162°
           7.12    ────────────────────────────
                        tan 252° . cos ( ─ 72°)  .  sin ( ─ 72°)
  
  
                             sin (180° ─ 72°) . tan 72° . sin (90° + 72°)
                      =  ────────────────────────────
                            tan (180° + 72°) . (cos 72°) . ( ─ sin 72°)
  
  
                                sin 72° . tan 72° . cos 72°
                      =  ─ ─────────────────────────
                                tan 72° . cos 72° . sin 72°
  
  
                      =  ─ 1                                                                                                                                   [ 7. ]