Question

The value of sec²(tan^-12) + cosec² (cot^-1 3) is ____________________.

Answer 1

We know

${\tan }^{-1}x={\sec }^{-1}\sqrt{1+x^2}$ and ${\cot }^{-1}x={\mathrm{cosec}}^{-1}\sqrt{1+x^2}$

So,

${\tan }^{-1}2={\sec }^{-1}\sqrt{1+2^2}={\sec }^{-1}\sqrt{5}$

${\cot }^{-1}3={\mathrm{cosec}}^{-1}\sqrt{1+3^2}={\mathrm{cosec}}^{-1}\sqrt{10}$

$\therefore {\sec }^2\left({\tan }^{-1}2\right)+{\mathrm{cosec}}^2\left({\cot }^{-1}3\right)$

$={\sec }^2\left({\sec }^{-1}\sqrt{5}\right)+{\mathrm{cosec}}^2\left({\mathrm{cosec}}^{-1}\sqrt{10}\right)$

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

$={\left(\sqrt{5}\right)}^2+{\left(\sqrt{10}\right)}^2$

$=15$

Thus, the value of ${\sec }^2\left({\tan }^{-1}2\right)+{\mathrm{cosec}}^2\left({\cot }^{-1}3\right)$ is 15.

The value of sec²(tan⁻¹2) + cosec²(cot⁻¹ 3) is ____15____.