Sums Over Sums

A fraction with a sum on top and a sum below often turns into two products that share a factor. Take $\frac{\sin\left(5x\right)+\sin\left(3x\right)}{\cos\left(5x\right)+\cos\left(3x\right)}$. The top is $2\sin\left(4x\right)\cos\left(x\right)$ and the bottom is $2\cos\left(4x\right)\cos\left(x\right)$. The 2 and the cos x cancel, and $\frac{\sin\left(4x\right)}{\cos\left(4x\right)}$ is $\tan\left(4x\right)$.

Turn both into products first, then look for the common factor. A sum cannot be cancelled term by term; a product can.