Answer
$$\sin^2\frac{3\pi}{8}=\frac{2+\sqrt2}{4}$$
Work Step by Step
$$\sin^2\frac{3\pi}{8}$$
*Recall the half-angle formula for sine, which is $$\sin^2\theta=\frac{1-\cos2\theta}{2}$$
Thus, $$\sin^2\frac{3\pi}{8}=\frac{1-\cos\frac{3\pi}{4}}{2}$$
Angle $\frac{3\pi}{4}$ is the equivalent of angle $\frac{\pi}{4}$ but in the second quadrant in the unit circle, where cosine is negative.
Therefore, $\cos\frac{3\pi}{4}=-\cos\frac{\pi}{4}=-\frac{\sqrt2}{2}$
So, continuing with the given function:
$$\sin^2\frac{3\pi}{8}=\frac{1-\Big(-\frac{\sqrt2}{2}\Big)}{2}$$
$$\sin^2\frac{3\pi}{8}=\frac{1+\frac{\sqrt2}{2}}{2}$$
$$\sin^2\frac{3\pi}{8}=\frac{\frac{2+\sqrt2}{2}}{2}$$
$$\sin^2\frac{3\pi}{8}=\frac{2+\sqrt2}{4}$$
