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

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

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.