Browse Questions

# The value of the expression $\large\frac{\sin^3x}{1+\cos x}+\frac{\cos^3x}{1-\sin x}$ is

$\begin{array}{1 1}(a)\;\sqrt 2\cos\big[\large\frac{\pi}{4}-\normalsize x\big]&(b)\;\sqrt 2\cos\big[\large\frac{\pi}{4}+\normalsize x\big]\\(c)\;\sqrt 2\sin\big[\large\frac{\pi}{4}-\normalsize x\big]&(d)\;None\;of\;these\end{array}$

Let $\large\frac{\sin^3x}{1+\cos x}+\frac{\cos^3x}{1-\sin x}$$=A Then A=\large\frac{(\sin^3x+\cos^3x)+(\cos^4x-\sin^4x)}{(1+\cos x)(1-\sin x)} A=\large\frac{(\sin^3x+\cos^3x)\big((\cos x+\sin x)(\cos x-\sin x)(\cos^2x+\sin^2x)\big)}{(1+\cos x)(1-\sin x)} A=\large\frac{(\sin x+\cos x)\{(1-\sin x\cos x)+(\cos x-\sin x)}{1+\cos x-\sin x-\sin x\cos x} Step 2: A=\sin x+\cos x A=\sqrt 2\big[\large\frac{1}{\sqrt 2}$$\sin x+\large\frac{1}{\sqrt 2}$$\cos x\big]------(1) A=\sqrt 2\big[\cos\large\frac{\pi}{4}$$\sin x+\sin\large\frac{\pi}{4}$$\cos x\big] \Rightarrow \sqrt 2\sin\big[\large\frac{\pi}{4}$$+x]$
$A=\sqrt 2\big[\sin\large\frac{\pi}{4}$$\sin x+\cos\large\frac{\pi}{4}$$\cos x\big]$