# How do you simplify sec(tan^(-1)(x))?

We have to evaluate $\sec (tan^{-1}x)$

### Solution

$\sec (tan^{-1}x)$

Let us assume that

$y=(tan^{-1}x)$

x = tan y

x = sin y / cos y

$x^{2}=\frac{(sin y)^{2}}{\(\cos y)^{2}}$ $x^{2}+ 1=\frac{\cos ^{2}y+ \sin ^{2}y}{\cos ^{2}y}$ $x^{2}+ 1=\sec ^{2}y$ $\sqrt{x^{2}+ 1}=\sec ^{2}y$ $\sqrt{x^{2}+ 1}=\sec (\tan ^{-1}x)$ $\sec (\tan ^{-1}x)=\sqrt{x^{2}+ 1}$

$\sec (\tan ^{-1}x)=\sqrt{x^{2}+ 1}$