If for some α, β ; α leq β, α+β=8 and sec 2 (tan -1 α )+ cosec 2 (cot -1 β )=36 , then α 2 +β is \ \ \ \ \ .

Mathematics · Inverse Trigonometric Functions

JEE Main 2025 — 24 January, Morning Shift — Question 22

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

Numerical answer — enter this value.

Step-by-step solution

$1 +\left(\tan \left(\tan ^{-1}(\alpha)\right)^{2}+1 +\left(\cot \left(\cot ^{-1} \beta\right)\right)^{2}=36\right.$

$\alpha^{2}+\beta^{2}=34$

$\alpha \beta=15$

$\alpha=3, \beta=5$

$\therefore \alpha^{2}+\beta=9+5=14$

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 24 January, Morning Shift
Subject: Mathematics
Chapter: Inverse Trigonometric Functions
Topic: Equations, Inequations and Identitites involving ITFs