Answer
$\frac{\sqrt 2+\sqrt 6}{4}$
Work Step by Step
Step 1. In order to use the Subtraction Formula, we need to think of two values whose difference is $\frac{5\pi}{12}$
Step 2. Notice that $\frac{5\pi}{12}=\frac{(9-4)\pi}{12}=\frac{9\pi}{12}-\frac{4\pi}{12}=\frac{3\pi}{4}-\frac{\pi}{3}$, we have $sin\frac{5\pi}{12}=sin(\frac{3\pi}{4}-\frac{\pi}{3})=sin\frac{3\pi}{4}cos\frac{\pi}{3}-cos\frac{3\pi}{4}sin\frac{\pi}{3}
=\frac{\sqrt 2}{2}\times\frac{1}{2}+\frac{\sqrt 2}{2}\times\frac{\sqrt 3}{2}=\frac{\sqrt 2+\sqrt 6}{4}$
