The values of sec2⁡tan−1⁡2+cosec2⁡cot−1⁡3=

# The values of ${\mathrm{sec}}^{2}\left({\mathrm{tan}}^{-1}2\right)+{\mathrm{cosec}}^{2}\left({\mathrm{cot}}^{-1}3\right)=$

1. A

5

2. B

10

3. C

15

4. D

20

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### Solution:

We have,

$\begin{array}{l}{\mathrm{sec}}^{2}\left({\mathrm{tan}}^{-1}2\right)+{\mathrm{cosec}}^{2}\left({\mathrm{cot}}^{-1}3\right)\\ ={\left\{\mathrm{sec}\left({\mathrm{tan}}^{-1}2\right)\right\}}^{2}+{\left\{\mathrm{cosec}\left({\mathrm{cot}}^{-1}3\right)\right\}}^{2}\\ ={\left\{\mathrm{sec}\left({\mathrm{tan}}^{-1}\frac{2}{1}\right)\right\}}^{2}+{\left\{\mathrm{cosec}\left({\mathrm{cot}}^{-1}\frac{3}{1}\right)\right\}}^{2}\\ ={\left\{\mathrm{sec}\left({\mathrm{sec}}^{-1}\sqrt{5}\right)\right\}}^{2}+{\left\{\mathrm{cosec}\left({\mathrm{cosec}}^{-1}\sqrt{10}\right)\right\}}^{2}\\ =\left(\sqrt{5}{\right)}^{2}+\left(\sqrt{10}{\right)}^{2}=15\end{array}$

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