JEE PYQ: Inverse Trigonometric Functions Question 8
Question 8 - 2021 (25 Feb Shift 2)
$\operatorname{cosec}\left[2\cot^{-1}(5) + \cos^{-1}\left(\frac{4}{5}\right)\right]$ is equal to:
(1) $\frac{75}{56}$
(2) $\frac{65}{56}$
(3) $\frac{56}{33}$
(4) $\frac{65}{33}$
Show Answer
Answer: (2)
Solution
$\operatorname{cosec}\left(2\tan^{-1}\frac{1}{5} + \cos^{-1}\frac{4}{5}\right)$. Let $\tan^{-1}\frac{5}{12} = \phi_1$ (from $2\tan^{-1}\frac{1}{5} = \tan^{-1}\frac{5}{12}$) and $\cos^{-1}\frac{4}{5} = \phi_2$, so $\sin\phi_2 = \frac{3}{5}$. Then $\operatorname{cosec}(\phi_1 + \phi_2) = \frac{1}{\sin\phi_1\cos\phi_2 + \cos\phi_1\sin\phi_2} = \frac{1}{\frac{5}{13}\cdot\frac{4}{5} + \frac{12}{13}\cdot\frac{3}{5}} = \frac{1}{\frac{56}{65}} = \frac{65}{56}$.
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