Prove that: x+ 3 x x+ 3 x = 2 x

Question

Prove that: $\frac{\sin x+\sin 3 x}{\cos x+\cos 3 x}=\tan 2 x$

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Answer

To prove: $\frac{\sin x+\sin 3 x}{\cos x+\cos 3 x}=\tan 2 x$
Taking L.H.S, we have
L.H.S = $\frac{\sin x+\sin 3 x}{\cos x+\cos 3 x}$
We know,
cos A + cos B = 2 cos $\frac{(A+B)}{2} \cos \frac{(A-B)}{2}$& sin A + sin B = 2 sin$\frac{(A+B)}{2} \cos \frac{(A-B)}{2}$
LHS = $\frac{2 \sin \frac{(x+3 x)}{2} \cos \frac{(x-3 x)}{2}}{2 \cos \frac{(x+3 x)}{2} \cos \frac{(x-3 x)}{2}}$ = $\frac{\sin 2 x \cos (-x)}{\cos 2 x \cos (-x)}=\frac{\sin 2 x}{\cos 2 x}$
LHS = tan 2x
$\therefore$ LHS = RHS
Hence, proved.

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