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= 2 sin 8θ cos 4θ
= 2 sin 2θ cos 3θ
= 2cos 10θ cos 2θ
(iv) cos 12θ - cos 4θ
= -2 sin 8θ sin 4θ
(v) sin 2θ + cos 4θ
= sin 2θ + sin (90 - 4θ)
sin 38° + sin 22° = sin 82°
LHS = sin 38°+ sin 22°
= 2 sin 30°cos 8°
= cos(90 - 82)°
= sin 82° = RHS [∵ cos θ = sin (90 - θ)]
cos 100° + cos 20° = cos 40°
= 2cos 60°cos 40°
sin 50°+ sin 10° = cos 20°
= 2 sin 30°cos 20°
Sin 30° + sin 37° = cos 7°
LHS = sin 23° + sin 37°
= 2 sin (30°) cos (-7°)
= cos 7° = RHS
LHS = sin 105° + cos 105°
= sin 105° + cos (90° + 15°)
= sin 105° - sin 15°
= 2 sin 45°cos 60°
= cos 45°
cos 80° + cos 40° - cos 20° = 0
(cos 80° + cos 40°) - cos 20°
= 2cos 60°cos 20°- cos 20°
= cos 20° - cos 20°
= 0
= RHS
cos 20° + cos100° + cos140° = 0
⟹ (cos 20° + cos100°) + cos140°
= 2cos 60°cos (- 40°) + cos140°
= cos 40° + cos (180° - 40°)
= cos 40° - cos 40°
= sin 50°- cos 80°
= sin 50°- sin 10°
= 2 sin 20° cos 30°
Multiplying and dividing by √2 on LHS
sin 80° - cos 70° = cos 50°
LHS = sin 80° = cos 50° + cos 70°
Now,
RHS = cos 50° + cos 70°
= 2 cos 60°cos (-10°)
= cos 10°
= sin 80°
= LHS [∵ cos θ = sin (90 - θ)]
sin 51° + cos 81° = cos 21°
sin 51° = cos 21° - cos 81°
RHS = cos 21° – cos 81°
= sin 51°
LHS
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Chapter 8: Transformation Formulae –...