Prove that: x- 3 x ^2 x- ^2 x = 2 x - Vidyadip

Question

Prove that: $\frac{\sin x-\sin 3 x}{\sin ^2 x-\cos ^2 x} = 2\sin x$

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Answer

We have,
$ \text { L. H. S.} =\frac{\sin x-\sin 3 x}{\sin ^2 x-\cos ^2 x}=\frac{-(\sin 3 x-\sin x)}{-\left(\cos ^2 x-\sin ^2 x\right)}=\frac{\left(\sin 3 x-\sin ^2 x\right)}{\left(\cos ^2 x-\sin ^2 x\right)}$
$=\frac{2 \cos \left(\frac{3 x+x}{2}\right) \sin \left(\frac{3 x-x}{2}\right)}{\cos 2 x}$
$=\frac{2 \cos 2 x \sin x}{\cos 2 x}=2 \sin x $

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