Try doubling the angle, and then turning around and using half-angle formulas for the tangent.
This is an exercise in Trig identities. First notice that 90 - 67.5 = 22.5 which is half of 45. In degrees, tan(90-x) = cot(x). So it is enough to calculate tan(22.5). Then use half-angle. You have tan(22.5) = (1-cos(45))/sin(45) .