Q62 / 160 · TS EAMCET 2021 (06 Aug Shift 2) · Mathematics

If $A$ is not an integral multiple of $\frac{\pi}{2}$, then $\operatorname{cosec} 2 A+\cot 2 A$ is equal to

  • A
    tan A
  • B
    $\cot A+2 \cot 2 A$
  • Correct answer
    C
    $\tan A+2 \cot 2 A$
  • D
    tan2 A

Solution

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

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

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