A) \[\sqrt{2}\cos \left( \frac{\pi }{4}-x \right)\]
B) \[\sqrt{2}\cos \left( \frac{\pi }{4}+x \right)\]
C) \[\sqrt{2}\sin \left( \frac{\pi }{4}-x \right)\]
D) None of these
Correct Answer: A
Solution :
Let \[A=\frac{{{\sin }^{3}}x}{1+\cos x}+\frac{{{\cos }^{3}}x}{1-\sin x}\] \[=\frac{({{\sin }^{3}}x+{{\cos }^{3}}x)+({{\cos }^{4}}x-{{\sin }^{4}}x)}{(1+\cos x)(1-\sin x)}\] \[=\frac{\begin{align} & [(\sin x+\cos x)(1-\sin x.\cos x)] \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+[(\cos x+\sin x)(\cos x-\sin x)] \\ \end{align}}{(1+\cos x)(1-\sin x)}\] \[=\frac{(\sin x+\cos x)[(1-\sin x.\cos x)(\cos x-\sin x)]}{1+\cos x-\sin x-\sin x.\cos x}\] \[=\sin x+\cos x\] \[=\sqrt{2}\left\{ \frac{1}{\sqrt{2}}.\sin x+\frac{1}{\sqrt{2}}.\cos x \right\}\] \[=\sqrt{2}\left\{ \cos \frac{\pi }{4}.\sin x+\sin \frac{\pi }{4}.\cos x \right\}\] \[=\sqrt{2}\sin \left( \frac{\pi }{4}+x \right)\] \[=\sqrt{2}\cos \left( \frac{\pi }{4}-x \right)\]
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