# How do you simplify Cos[tan^-1(-1)]?

Jun 26, 2016

$\frac{1}{\sqrt{2}}$, for the principal value $- \frac{\pi}{4}$ of ${\tan}^{- 1} \left(- 1\right)$.
Against , $\frac{3 \pi}{4}$, the answer is $- \frac{1}{\sqrt{2}}$.

#### Explanation:

If $a = {\tan}^{- 1} \left(- 1\right) , \tan a = - 1.$,

a is in either 4th quadrant with the principal value $- \frac{\pi}{4}$ or in

the 2nd, as $\frac{3 \pi}{4}$.

So, the given cosine cos a is

$\cos \left(- \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$ or

$\cos \left(\frac{3 \pi}{4}\right) = \cos \left(\pi - \frac{\pi}{4}\right) = - \cos \left(\frac{\pi}{4}\right) = - \frac{1}{\sqrt{2}}$.

In brief, the answer is $\pm \frac{1}{\sqrt{2}}$ for the general value of ${\tan}^{- 1} \left(- 1\right)$..