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  4. Find the value of sec2(tan-12) + cosec2(cot-13)

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Q. Find the value of $sec^2(tan^{-1}2) + cosec^2(cot^{-1}3)$

Inverse Trigonometric Functions

Solution:

$sec^2(tan^{-1}2) + cosec^2(cot^{-1}3)$
$= \left[sec\left(tan^{-1}2\right)\right]^{2}+ \left[cosec\left(cot^{-1}3\right)\right]^{2}$
$= \left[sec\left(sec^{-1}\sqrt{5}\right)\right]^{2}+\left[cosec\left(cosec^{-1}\sqrt{10}\right)\right]^{2}$
$= \left(\sqrt{5}\right)^{2} + \left(\sqrt{10}\right)^{2}= 5 + 10 = 15$