[Math] How to simplify a radical within a radical in this half-angle problem

Question

I don't understand how to simplify the following radicals and arrive at the final answer below. I can make it to this point:

$$\sin\left(-\frac{3\pi}{8}\right)=\pm\sqrt{1+\frac{\sqrt2}{2}\over2}$$

However the final answer is:

$$\sin\left(-\frac{3\pi}{8}\right)=-\frac{\sqrt{2+\sqrt{2}}}{2}$$

I've filled a couple pages and tried finding a good answer on how to do this simplification, but without success. Any help is gratefully received!

edit: removed the square over the denominator 2 in the final answer.

Answer 1

First, $\;-\frac\pi2<-\frac{3\pi}8< 0\;$ , so we're in the fourth quadrant and thus sine is negative here. Second:

$$\sqrt{\frac{1+\frac{\sqrt2}{2}}{2}}=\sqrt{\frac{\frac{2+\sqrt2}2}{2}}=\sqrt{\frac{2+\sqrt2}4}=\frac{\sqrt{2+\sqrt2}}2$$