How do you prove #sin 5pi/12 = sin pi/6 cos pi/4 + cos pi/6 sin pi/4#?

You rewrite 5π/12 as  2π/12 + 3π/12 since 2 + 3 = 5.
Then you simplify the fractions into π/6 + π/4.
Then you use the sine addition identity.

sin(5π/12) = sin(π/6 + π/4) = sin π/6 cos π/4 + cos π/6 sin π/4