arcsec(-sqrt(2)) Evaluate the expression without using a calculator

For a negative number x (actually, for x less than or equal to -1 ) its arcsecant is such number y in (pi/2, pi] that  sec(y) = x. This number is unique.
This is the same as cos(y) = 1/x, for the given x = -sqrt(2) it is -1/sqrt(2). We know such angle, it is  (3 pi)/4.
So the answer is  (3 pi)/4.