Verified Solution • 2025–26 Session moderate

QJEEM25Ma0fac: JEE Main 2025 Mathematics - Class 12 JEEM25Ma0fac Solution: If for some and , then is __________

If for some α,β;α≤β,α+β=8\alpha, \beta ; \alpha \leq \beta, \alpha+\beta=8 and sec⁡2(tan⁡−1α)+cosec⁡2(cot⁡−1β)=36\sec ^2\left(\tan ^{-1} \alpha\right)+\operatorname{cosec}^2\left(\cot ^{-1} \beta\right)=36 , then α2+β\alpha^2+\beta is __________

Step-by-Step Verified Solution

amp;If⁡(tan⁡(tan⁡−1(α))+1(cot⁡(cot⁡−1β))2=36amp;α2+β2=34amp;αβ=15amp;α=3,β=5amp;∴α2+β=9+5=14\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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