Q67 / 180 · COMEDK 2017 · Mathematics

The value of $2 \operatorname{cosec} 2 x+\operatorname{cosec} x$ is equal to

  • A
    $\tan x \cdot \sec (x / 2) \quad$
  • Correct answer
    B
    $\sec x \cdot \cot (x / 2)$
  • C
    $\sec x \cdot \tan (x / 2) \quad$
  • D
    $\tan x \cdot \cot (x / 2)$

Solution

$2 \operatorname{cosec} 2 x+\operatorname{cosec} x$ $\begin{aligned} &=\frac{2}{\sin 2 x}+\frac{1}{\sin x}=\frac{2}{2 \sin x \cdot \cos x}+\frac{1}{\sin x} \\ &=\frac{1}{\sin x}\left[\frac{1}{\cos x}+1\right]=\frac{1}{\sin x} \cdot \frac{(1+\cos x)}{\cos x} \\ &=\frac{1}{\cos x} \cdot \frac{2 \cos ^{2}(x / 2)}{2 \cdot \sin (x / 2) \cdot \cos (x / 2)} \\ &=\sec x \cdot \frac{\cos (x / 2)}{\sin (x / 2)} \quad\left[\begin{array}{l} \because \cos 2 x=2 \cos ^{2} x-1 \\ \sin 2 x=2 \sin x \cos x \end{array}\right] \\ &=\sec x \cdot \cot (x / 2) \quad \end{aligned}$

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

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