# Find the value of tan(cos^(-1)(3/x))?

Oct 24, 2017

$\tan \left({\cos}^{- 1} \left(\frac{3}{x}\right)\right) = \tan t = \frac{1}{3} \sqrt{{x}^{2} - 9}$

#### Explanation:

Let ${\cos}^{- 1} \left(\frac{3}{x}\right) = t$

then $\cos t = \frac{3}{x}$

and therefore $\sec t = \frac{x}{3}$

and ${\sec}^{2} x = {x}^{2} / 9$ and ${\tan}^{2} t = {x}^{2} / 9 - 1 = \frac{{x}^{2} - 9}{9}$

and $\tan t = \frac{1}{3} \sqrt{{x}^{2} - 9}$

hence $\tan \left({\cos}^{- 1} \left(\frac{3}{x}\right)\right) = \tan t = \frac{1}{3} \sqrt{{x}^{2} - 9}$