Prove that 1 - cos 2 x + sin x sin 2 x + cos x = tan x .

To prove that 1 - cos 2 x + sin x sin 2 x + cos x = tan x . Now, 1 - cos 2 x + sin x sin 2 x + cos x = 2 sin 2 x + sin x 2 sin x cos x + cos x ∵ cos 2 x = 1 - 2 sin 2 x = sin x ( 2 sin x + 1 ) cos x ( 2 sin x + 1 ) = sin x cos x = tan x Hence, proved.

Prove that 1 - cos 2 x + sin x sin 2 x + cos x = tan x .

To prove that 1 - cos 2 x + sin x sin 2 x + cos x = tan x . Now, 1 - cos 2 x + sin x sin 2 x + cos x = 2 sin 2 x + sin x 2 sin x cos x + cos x ∵ cos 2 x = 1 - 2 sin 2 x = sin x ( 2 sin x + 1 ) cos x ( 2 sin x + 1 ) = sin x cos x = tan x Hence, proved.

Prove that cos x cos 2 x cos 4 x cos 8 x = sin 16 x 16 sin x .

To prove that cos x cos 2 x cos 4 x cos 8 x = sin 16 x 16 sin x . Now, cos x cos 2 x cos 4 x cos 8 x = 2 sin x 2 sin x cos x cos 2 x cos 4 x cos 8 x [Multiply and divide by 2 sin x ] = 2 sin 2 x 4 sin x cos 2 x cos 4 x cos 8 x ∵ sin 2 A = 2 sin A cos A = 2 sin 4 x 8 sin x cos 4 x cos 8 x = 2 sin 8 x 16 sin x cos 8 x = sin 16 x 16 sin x Hence, proved.

Prove that 2 sin 22 1 2 ° cos 22 1 2 ° = 1 2 .

To prove that 2 sin 22 1 2 ° cos 22 1 2 ° = 1 2 Now, 2 sin 22 1 2 ° cos 22 1 2 ° = sin 2 22 1 2 ° ∵ 2 sin x cos x = sin 2 x = sin 45 ° = 1 2 Hence, proved.

Prove that 2 cos 2 15 ° - 1 = 3 2 .

To prove that 2 cos 2 15 ° - 1 = 3 2 . Now, 2 cos 2 15 ° - 1 = cos 2 15 ° ∵ cos 2 θ = 2 cos 2 θ - 1 = cos 30 ° = 3 2 Hence, proved.

Prove that 3 sin 40 ° - 4 sin 3 40 ° = 3 2 .

To prove that 3 sin 40 ° - 4 sin 3 40 ° = 3 2 . Now, 3 sin 40 ° - 4 sin 3 40 ° = sin 3 40 ° ∵ sin 3 x = 3 sin x - 4 sin 3 x = sin 120 ° = sin 180 ° - 60 ° = sin 60 ° ∵ sin ( π - θ ) = sin θ = 3 2 Hence, proved.

Prove that sin 2 24 ° - sin 2 6 ° = 5 - 1 8 .

To prove, sin 2 24 ° - sin 2 6 ° = 5 - 1 8 Now, sin 2 24 ° - sin 2 6 ° = sin 24 ° + 6 ° sin 24 ° - 6 ° ∵ sin 2 x - sin 2 y = sin ( x - y ) sin ( x + y ) = sin 30 ° sin 18 ° = 1 2 · 5 - 1 4 = 5 - 1 8 Hence, proved.

Prove that sin 2 72 ° - cos 2 30 ° = 5 - 1 8 .

To prove sin 2 72 ° - cos 2 30 ° = 5 - 1 8 Now, sin 2 72 ° - cos 2 30 ° = cos 2 18 ° - 3 2 2 ∵ sin 72 ° = sin 90 ° - 18 ° = cos 18 ° = 1 - sin 2 18 ° - 3 4 ∵ sin 2 x + cos 2 x = 1 = 1 4 - 5 - 1 4 2 = 4 16 - ( 5 + 1 - 2 5 ) 16 = - 2 + 2 5 16 = 5 - 1 8 Hence, proved.