Please show the following trigonometric identity.?

Question

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Answer 1

We have 2 cos A sin B = sin (A+B) – sin (A-B)


So

2 Cos x sin x/2 = sin 3x/2 – sin x/2
2 Cos 2x sin x/2 = sin 5x/2- sin 3x/2

2 Cos nx sin x/2 = sin (n+1/2) x – sin (n-1/2) x

Adding we get 2 sin x/2( cos x + cos 2x ----- + cos (nx) = sin (n+1/2) x – sin x/2

Dividing both sides by 2sin x/2 we get the result