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

Question

4404 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-06-04T22:22:31

Call tan 67.5 = t. Use trig identity: $tan 2 a = \frac{2 tan a}{1 - {tan}^{2} a}$ $tan 2a = (2tan a)/(1 - tan^2 a)$

$tan 135 = tan (45 + 90) = cot 45 = 1 = \frac{2 t}{1 - t^{2}}$ $tan 135 = tan (45 + 90) = cot 45 = 1 = (2t)/(1 - t^2)$

$f (t) = t^{2} + 2 t - 1 = 0$ $f(t) = t^2 + 2t - 1 = 0$

$D = d^{2} = 4 + 4 = 8 \to d = \pm 2 \sqrt{2}$ $D = d^2 = 4 + 4 = 8 -> d = +- 2sqrt2$

$t = - 1 \pm \sqrt{2}$ $t = -1 +- sqrt2$

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

How do you use a half-angle formula to simplify tan67.5tan 67.5?