Question

How do you use a half-angle formula to simplify `tan 67.5`?

Answer 1

Call tan 67.5 = t. Use trig identity: `tan 2a = (2tan a)/(1 - tan^2 a)`

`tan 135 = tan (45 + 90) = cot 45 = 1 = (2t)/(1 - t^2)`

`f(t) = t^2 + 2t - 1 = 0`

`D = d^2 = 4 + 4 = 8 -> d = +- 2sqrt2`

`t = -1 +- sqrt2`

tan 67.5 = sqrt2 - 1 = 0.414 (Quadrant I)