Denest $\sqrt{3+\sin\frac{\pi}{4}-4\cos\frac{\pi}{8}}$ in terms of possible trigonometric functions.

Question

As the title suggests, denest; $$\sqrt{3+\sin\frac{\pi}{4}-4\cos\frac{\pi}{8}}$$ in terms of possible trigonometric functions.

Answer 1

Let $c:=\cos\frac{\pi}{8}$ so your surd is $\sqrt{3+\cos\frac{\pi}{4}-4c}=\sqrt{2c^2-4c+2}=(1-c)\sqrt{2}$, which you can freel to rewrite e.g. as $\sqrt{8}\sin^2\frac{\pi}{16}$. But you can rewrite it without trigonometry at all, since $c=\sqrt{\frac{1+1/\sqrt{2}}{2}}$.