Question

Use a half-angle identity to find the exact solution of cos(3pi/8).

Answer 1

If $0$, then $\cos x>0$.

Recall the double angle identity,

$\cos^2x=\dfrac{1+\cos(2x)}2$

so we have

$\cos^2\dfrac{3\pi}8=\dfrac{1+\cos\frac{6\pi}8}2=\dfrac{1-\frac1{\sqrt2}}2=\dfrac{2-\sqrt2}4$

$\implies\cos\dfrac{3\pi}8=\boxed{\dfrac{\sqrt{2-\sqrt2}}2}$