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.