Answer
$$\sin\frac{7\pi}{12}=\frac{\sqrt2+\sqrt6}{4}$$
Work Step by Step
As suggested by the exercise, we would rewrite:
$$\frac{7\pi}{12}=\frac{\pi}{4}+\frac{\pi}{3}$$
Therefore, $$\sin\frac{7\pi}{12}=\sin\Big(\frac{\pi}{4}+\frac{\pi}{3}\Big)$$
Apply the addition formula for sine here:
$$\sin\frac{7\pi}{12}=\sin\frac{\pi}{4}\cos\frac{\pi}{3}+\cos\frac{\pi}{4}\sin\frac{\pi}{3}$$
Remember that $\sin\frac{\pi}{4}=\cos\frac{\pi}{4}=\frac{\sqrt2}{2}$ and $\sin\frac{\pi}{3}=\frac{\sqrt3}{2}$ and $\cos\frac{\pi}{3}=\frac{1}{2}$
$$\sin\frac{7\pi}{12}=\frac{\sqrt2}{2}\times\frac{1}{2}+\frac{\sqrt2}{2}\times\frac{\sqrt3}{2}$$
$$\sin\frac{7\pi}{12}=\frac{\sqrt2}{4}+\frac{\sqrt6}{4}$$
$$\sin\frac{7\pi}{12}=\frac{\sqrt2+\sqrt6}{4}$$
