Question

Prove the identity
(cos 3x+cos x)/(sin3x-sinx)=cotx

Answer 1

\[\cos x+\cos y=2\cos \frac{x+y}{2}\cos \frac{x-y}{2}\]

Answer 2

So cos 3x+cosx=2cos2xcosx
Also:
\[\sin x+\sin y=2\cos \frac{x+y}{2}\cos \frac{x-y}{2}\]

Answer 3

So sin3x-sinx = 2cos2xsinx

Answer 4

So
\[\frac{\cos 3x+\cos x}{\sin 3x-\sin x}=\frac{2\cos 2x \cos x}{2\cos 2x \sin x}=\frac{\cos x}{\sin x}=\cot x\]

Answer 5

\[\sin x+\sin y=2\sin \frac{x+y}{2}\cos \frac{x-y}{2} ****\]