JEE Main 2025 (Online) 24th January Morning ShiftQuestion 1
If for some $\alpha,\beta$; $\alpha\le\beta$, $\alpha+\beta=8$ and $\sec^{2}\left(\tan^{-1}\alpha\right)+\operatorname{cosec}^{2}\left(\cot^{-1}\beta\right)=36$, then $\alpha^{2}+\beta$ is
Inverse Trigonometric Functions
Answer$14$
Solution
$\sec^2(\tan^{-1}\alpha)=1+\alpha^2$, $\operatorname{cosec}^2(\cot^{-1}\beta)=1+\beta^2$, so $\alpha^2+\beta^2=34$, $\alpha+\beta=8$, giving $\alpha=3$, $\beta=5$. $\alpha^2+\beta=14$.