Products to Sums

Add sin(A + B) and sin(A - B): the cos A sin B terms cancel and two equal products are left. That, and its two cousins, turn a product into a sum:

$$2\sin\left(A\right)\cos\left(B\right) = \sin\left(A+B\right)+\sin\left(A-B\right)$$

$$2\cos\left(A\right)\cos\left(B\right) = \cos\left(A-B\right)+\cos\left(A+B\right)$$

$$2\sin\left(A\right)\sin\left(B\right) = \cos\left(A-B\right)-\cos\left(A+B\right)$$

With A = 3x and B = x, $2\sin\left(3x\right)\cos\left(x\right)$ is $\sin\left(4x\right)+\sin\left(2x\right)$. A sum is easier to graph, to add up, and in calculus to integrate.