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JEE Main · 2025 · 24 Jan 2025 · Shift 1 · Inverse Trigonometric Functions

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 ____.

Answer: 14

Solution

Using

sec^{2} (tan^{- 1} α) = 1 + α^{2}

and

csc^{2} (cot^{- 1} β) = 1 + β^{2}

:

(1 + α^{2}) + (1 + β^{2}) = 36 \Rightarrow α^{2} + β^{2} = 34.

Given

α + β = 8

:

(α + β)^{2} = α^{2} + 2 α β + β^{2} = 64 \Rightarrow 34 + 2 α β = 64 \Rightarrow α β = 15.

α, β

are roots of

t^{2} - 8 t + 15 = 0

, i.e.,

(t - 3) (t - 5) = 0

, so

{α, β} = {3, 5}

. Since

α \leq β

,

α = 3, β = 5

.

α^{2} + β = 9 + 5 = 14.

JEE Main · 2025 · 24 Jan 2025 · Shift 1 · Integer answer

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