What significance do squares of trigonometric ratios have algebraically and geometrically? I recently came upon a problem which asked me to:
"Show that $(\cos x + 1)^2 = \frac{\cos2x}{2} + 2\cos x + \frac32$ and hence determine $\int(\cos x + 1)^2 dx$."; 
from Mathematics for the International Student-IB Diploma: SL. Being given a value with which to substitute the argument of the integral meant to be found, the second task proved manageable:
Since $(\cos x + 1)^2 = \frac{\cos2x}{2} + 2\cos x + \frac32$, it is evident that $\int(\cos x + 1)^2 dx = \int(\frac{\cos2x}{2} + 2\cos x + \frac32)dx$, which is solved quite easily. 
However, I struggled at finding a way to show that the initial argument made sense. Here is the work I did:
$$\frac{\cos 2x}{2} + 2 \cos x + \frac32 = (\cos x + 1)^2$$
$$\frac{\cos 2x}{2} + 2 \cos x + \frac32= \cos^2 x + 2 \cos x + 1$$
$$\frac{\cos 2x + 3}{2} = \cos^2 x + 1$$
$$\cos 2x + 3 = 2 \cos^2 x + 2$$
$$\cos 2x = 2\cos^2 x - 1$$
It is at this point that I get stuck. I realize that trigonometric identities will facilitate the above statement, but I do not know which ones "fit". Therefore, I come to my question: what significance do squares of trigonometric ratios have algebraically and geometrically?

Expanding the expression we have $$(\cos x + 1)^2 = \cos ^2 x + 2 \cos x + 1$$
However, $\cos^2 x = \frac {\cos 2x + 1}{2}$ so we have $$\frac {\cos 2x + 1}{2} + 2 \cos x + 1$$ Factoring $1/2$ from the first term $\frac {\cos 2x + 1}{2}$, we then have $$\frac {\cos 2x}{2} + \frac {1}{2} + 2 \cos x + 1 $$ or $$\frac {\cos 2x}{2} + 2 \cos x + \frac {3}{2}$$
The integral for $\int (\frac {\cos 2x}{2} + 2 \cos x + \frac {3}{2}) dx$ is $$\frac {\sin 2x}{4} + 2 \sin x + \frac {3x}{2} + C$$
