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  4. The value of the expression ( sin 3x/1+ cos x)+( cos 3x/1- sin x) is/are

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Q. The value of the expression $ \frac{{{\sin }^{3}}x}{1+\cos x}+\frac{{{\cos }^{3}}x}{1-\sin x} $ is/are

JamiaJamia 2013

Solution:

Let $ A=\frac{{{\sin }^{3}}x}{1+\cos x}+\frac{{{\cos }^{3}}x}{1-\sin x} $ $ =\frac{({{\sin }^{3}}x+{{\cos }^{3}}x)+({{\cos }^{4}}x-{{\sin }^{4}}x)}{(1+\cos x)(1-\sin x)} $ $ =\frac{\begin{align} & [(\sin x+\cos x)(1-\sin x.\cos x)] \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+[(\cos x+\sin x)(\cos x-\sin x)] \\ \end{align}}{(1+\cos x)(1-\sin x)} $ $ =\frac{(\sin x+\cos x)[(1-\sin x.\cos x)(\cos x-\sin x)]}{1+\cos x-\sin x-\sin x.\cos x} $ $ =\sin x+\cos x $ $ =\sqrt{2}\left\{ \frac{1}{\sqrt{2}}.\sin x+\frac{1}{\sqrt{2}}.\cos x \right\} $ $ =\sqrt{2}\left\{ \cos \frac{\pi }{4}.\sin x+\sin \frac{\pi }{4}.\cos x \right\} $ $ =\sqrt{2}\sin \left( \frac{\pi }{4}+x \right) $ $ =\sqrt{2}\cos \left( \frac{\pi }{4}-x \right) $