Question

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

Answer 1

`sqrt(x^2+2)/x`.

Explanation:

Let `a = arc tan (x/sqrt 2) in [-pi/2, pi/2}`, for principal value.

Then, `tan a = 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.