Double-Angle, Power-Reducing, and Half-Angle Formulas Question?

Question

Use a half-angle formula to find the exact value of each expression.

Tan 7Pi/8

Answer 1

Use:(tanx)^2 = (1 - cos2x)/1 + cos2x)
(tan(7π/8))^2 = (1 - cos 7π/4)/(1 + cos 7π/4)
Second quadrant tangent is negative:
tan(7π/8) = -√[(1 - (√2)/2)/(1 + (√2)/2)]
Under the radical multiply numerator and denominator
by (1 - (√2)/2) and simplify:
tan(7π/8) = -√[(1 - √2 + 1/2)/(1 - 1/2)]
= -√[(3/2 - √2)/(1/2)]
= -√(3 - 2√2)