JEE MAIN - Mathematics (2025 - 24th January Morning Shift - No. 24)
If for some $\alpha, \beta ; \alpha \leq \beta, \alpha+\beta=8$ and $\sec ^2\left(\tan ^{-1} \alpha\right)+\operatorname{cosec}^2\left(\cot ^{-1} \beta\right)=36$, then $\alpha^2+\beta$ is __________
Answer
14
Explanation
$$\begin{aligned}
& \operatorname{If}\left(\tan \left(\tan ^{-1}(\alpha)\right)+1\left(\cot \left(\cot ^{-1} \beta\right)\right)^2=36\right. \\
& \alpha^2+\beta^2=34 \\
& \alpha \beta=15 \\
& \alpha=3, \beta=5 \\
& \therefore \alpha^2+\beta=9+5=14
\end{aligned}$$
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