Question

express the given sum or difference as a product of sines and/or cosines

sin x+sin3x

Answer 1

sin + sin = 2sin(1/2 sum)sin(1/2 difference)

Take theta_1 = 3x and theta_2 = x. Then half the sum is 2x and half the difference is x. So you get\[\sin x + \sin 3x =2\sin 2x \sin x\]

Answer 2

But sin(2x)=2sin(x)cos(x), so you can go further:\[2\sin 2x \sin x=2.2\sin x \cos x \sin x =4\sin^2 x \cos x\]