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  4. The value of sec2( tan-1 3)+cosec2( cot-1 2) is equal to

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Q. The value of $\sec^2(\tan^{-1}\,3)+cosec^2(\cot^{-1}\,2)$ is equal to

Inverse Trigonometric Functions

Solution:

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