How do you use a half-angle formula to simplify $tan (292.5)$ $tan(292.5)$ ?

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Answer 1 · 2015-06-15T19:03:57

Simplify tan (292.5)

Call x = tan 292.5
tan = 2(292.5) = tan 585 = (135 + 360) = tan (135) = - tan (45) = - 1

Use trig identity: $tan 2 x = \frac{2 tan x}{1 - {tan}^{2} x}$ $tan 2x = (2tan x)/(1 - tan^2 x)$

$- 1 = \frac{2 x}{1 - x^{2}}$ $- 1 = (2x)/(1 - x^2)$ -> $x^{2} - 2 x - 1 = 0$ $x^2 - 2x - 1 = 0$

$D = d^{3} = 4 + 4 = 8 \to d = \pm 2 \sqrt{2}$ $D = d^3 = 4 + 4 = 8 -> d = +- 2sqrt2$

$tan (292.5) = x = 1 - (\sqrt{2})$ $tan (292.5) = x = 1 - (sqrt2)$ (Quadrant IV)

How do you use a half-angle formula to simplify tan(292.5)tan(292.5)?