Power Reduction

Solve $\cos\left(2x\right) = 2{\cos\left(x\right)}^{2}-1$ for the square and you get a way to remove a power:

$${\cos\left(x\right)}^{2} = \frac{1+\cos\left(2x\right)}{2}$$

$${\sin\left(x\right)}^{2} = \frac{1-\cos\left(2x\right)}{2}$$

A square becomes a first power of a double angle, which is easier to integrate and, as the next lesson shows, to graph.

Take $2{\cos\left(x\right)}^{2}$. Replace the square: $\frac{2\left(1+\cos\left(2x\right)\right)}{2}$. Split the fraction into $\frac{1}{2}+\frac{\cos\left(2x\right)}{2}$, Expand the 2 in front, and the line reads $1+\cos\left(2x\right)$.