How to solve this equation

Question

 (sin x + sin 2x)*(2 cos x - 1 ) = sin 3x?

Answer 1

The trigometric equation is (sin x + sin 2x) * (2 cos x - 1) = sin 3x.

(sin x + 2 sin x cos x) * (2 cos x - 1) = sin 3x         [ ∵ sin 2x = 2 sin x cos x ]

sin x(2 cos x + 1)(2 cos x - 1) = sin 3x

sin x[ (2 cos x)²- 1] = sin 3x                               [ ∵ (a + b)(a - b) = a²- b² ]

sin x[ 4 cos² x - 1] = sin 3x

sin x[ 4 (1 - sin² x) - 1] = sin 3x                         [ ∵ cos² x + sin² x = 1 ]

sin x[ 4 - 4sin² x - 1] = sin 3x

sin x[ 3 - 4sin² x] = sin 3x

3sin x - 4sin³ x = sin 3x                                   [ ∵ 3sin x - 4sin³ x = sin 3x ]

sin 3x = sin 3x

sin 3x - sin 3x = 0

0 = 0.

The above statement is true.

So, all values of x are solutions.