If for some = 8 and sec 2 ( tan – 1 ) + cosec 2 ( cot – 1 ) = 36 , then 2 + is __________. [2025]

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Solution

Q.
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 __________. [2025]

Ans.
(14)

Given, $\sec^{2} (\tan^{- 1} α) + {cosec}^{2} (\cot^{- 1} β) = 36$

$∴ 1 + \tan^{2} (\tan^{- 1} α) + 1 + \cot^{2} (\cot^{- 1} β) = 36$

$\Rightarrow α^{2} + β^{2} = 34$

Given, $α + β = 8$ ... (i)

$\Rightarrow α^{2} + β^{2} + 2 α β = 64$

$\Rightarrow 2 α β = 30 \Rightarrow α β = 15$

$∴ β - α = \sqrt{{(α + β)}^{2} - 4 α β}$ [ $∵ β \geq α$ ]

$β - α = 2$ ... (ii)

From (i) and (ii), we get

$α = 3, β = 5$

$∴ α^{2} + β = 9 + 5 = 14$

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