Here are two very elementary examples.
It's not 100% different from your Cauchy's inequality example, but the fact that if X is a random variable, then $(\mathbb{E}X)^2\leq\mathbb{E}X^2$ is very useful and follows from the fact that the difference equals the variance of X.
The fact that $|\cos x|\leq 1$ and $|\sin x|\leq 1$ follows from the fact that $\cos^2x+\sin^2x=1$.