# How do you evaluate csc(arctan(x/sqrt2)) without a calculator?

Sep 17, 2016

$\frac{\sqrt{{x}^{2} + 2}}{x}$.

#### Explanation:

Let $a = a r c \tan \left(\frac{x}{\sqrt{2}}\right) \in \left[- \frac{\pi}{2} , \frac{\pi}{2}\right\}$, for principal value.

Then, $\tan a = \frac{x}{\sqrt{2}}$ and,

the given expression,

csc a = sqrt(1+cot^2 a)=sqrt(1+2/x^2)=sqrt(x^2+2)/x#.

Here x decides the sign of both tan a and csc a.