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

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Answer 1 · 2015-06-04T22:22:31

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)

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