Q61 / 160 · TS EAMCET 2020 (10 Sep Shift 1) · Mathematics
$\operatorname{cosec}^{-1}\left[\left(\frac{\tan ^2\left(\frac{\alpha-\pi}{4}\right)-1}{\tan ^2\left(\frac{\alpha-\pi}{4}\right)+1}+\cos \frac{\alpha}{2} \cdot \cot 5 \alpha\right)_{\sec \frac{11 \alpha}{2}}\right]$
- A$2 \alpha$
- Correct answerB$5 \alpha$
- C$\frac{\pi}{2}-4 \alpha$
- D$\frac{5}{2} \alpha$
Solution
We have,
$\begin{aligned}
& \operatorname{cosec}^{-1}\left[\left(\frac{\tan ^2\left(\frac{\alpha-\pi}{4}\right)-1}{\tan ^2\left(\frac{\alpha-\pi}{4}\right)+1}+\cos \frac{\alpha}{2} \cot 5 \alpha\right) \sec \frac{11 \alpha}{2}\right] \\
& =\operatorname{cosec}^{-1}\left[\left(-\cos 2\left(\frac{\alpha-\pi}{4}\right)+\cos \frac{\alpha}{2} \cot 5 \alpha\right) \sec \frac{11 \alpha}{2}\right] \\
& =\operatorname{cosec}^{-1}\left[\left(-\sin \frac{\alpha}{2}+\cos \frac{\alpha}{2} \frac{\cos 5 \alpha}{\sin 5 \alpha}\right) \sec \frac{11 \alpha}{2}\right] \\
& =\operatorname{cosec}^{-1}\left[\left(\frac{-\sin 5 \alpha \sin \frac{\alpha}{2}+\cos 5 \alpha \cos \frac{\alpha}{2}}{\sin 5 \alpha}\right) \sec \frac{11 \alpha}{2}\right] \\
&
\end{aligned}$
$\begin{aligned} & =\operatorname{cosec}^{-1}\left[\frac{\cos \left(5 \alpha+\frac{\alpha}{2}\right)}{\sin 5 \alpha} \cdot \sec \frac{11 \alpha}{2}\right] \\ & =\operatorname{cosec}^{-1}\left[\operatorname{cosec} 5 \alpha \cdot \cos \frac{11 \alpha}{2} \sec \frac{11 \alpha}{2}\right] \\ & =\operatorname{cosec}^{-1}(\operatorname{cosec} 5 \alpha)=5 \alpha\end{aligned}$
Hence, the correct answer is option: 'B'.