Q233 / 270 · NDA 2011 (Phase 2) · Mathematics

What is $\frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x}$ equal to?

  • A
    $2 \tan x$
  • Correct answer
    B
    $2 \operatorname{cosec} x$
  • C
    $2 \cos x$
  • D
    $2 \sin x$

Solution

$\begin{aligned} & \frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x} \\ & =\frac{\sin ^2 x+(1+\cos x)^2}{\sin x(1+\cos x)} \\ & =\frac{\sin ^2 x+1+2 \cos x+\cos ^2 x}{\sin x(1+\cos x)} \\ & =\frac{\left(\sin ^2 x+\cos ^2 x\right)+1+2 \cos x}{\sin x(1+\cos x)} \\ & =\frac{2+2 \cos x}{\sin x(1+\cos x)}=\frac{2(1+\cos x)}{\sin x(1+\cos x)} \\ & =2 \operatorname{cosec} x\end{aligned}$

Hence, the correct answer is option: 'B'.

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