Question

For which x is the complex cosine function real?

Answer 1

If e^(ix) = a+ib then e^(-ix) = 1/e^(ix) = (a-ib)/(a^(2)+b^(2))

Show you need a^(2)+b^(2) = 1 or b = 0.

What does that imply about x?

Answer 2

cos(a+bi) = cos(a)cosh(b) - i sin(a)sinh(b). you want the imaginary part to be 0, so you want sin(a) = 0 or sinh(b) = 0. so a = n pi or b = 0.