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

##### 2 Answers
Aug 2, 2018

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

#### Explanation:

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

$= \tan \left({\cos}^{- 1} \left(\frac{1}{x}\right)\right)$

let $y = {\cos}^{- 1} \left(\frac{1}{x}\right)$
$x = \frac{1}{\cos} \left(y\right)$

${x}^{2} = \frac{1}{\cos} {\left(y\right)}^{2}$

${x}^{2} - 1 = \frac{{\cancel{1 - \cos {\left(y\right)}^{2}}}^{= \sin {\left(y\right)}^{2}}}{\cos {\left(y\right)}^{2}}$

${x}^{2} - 1 = \tan {\left(y\right)}^{2}$

$\sqrt{{x}^{2} - 1} = \tan \left(y\right) = \tan \left({\sec}^{- 1} \left(x\right)\right)$

\0/ Here's our answer !

Aug 3, 2018

color(crimson)(tan(sec^-1(x)) =sqrt ( x^2 -1)

#### Explanation:

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

Let ${\sec}^{-} 1 x = y$

$x = \sec y$

${x}^{2} = {\sec}^{2} y$

x^2 = 1 + tan^2 y, color(brown)(sec^2 y = 1 + tan^2 y, " Identity"

${x}^{2} - 1 = {\tan}^{2} y$

$\tan y = \sqrt{{x}^{2} - 1}$

color(crimson)(tan(sec^-1(x)) = sqrt(x^2 -1)