How would I go about solving $\displaystyle |\cos x|^{1+\sin x+\cos x} \geq 1$? All help is welcome.
By THINKING, not using "formulas"!
You know that $\displaystyle |cos(x)|$ is always less than or equal to 1. You also know that powers of a number between 0 and 1 are always smaller than the number itself: if $\displaystyle 0< y< 1$ then $\displaystyle y^n< y$.
Saying that an absolute value of a power of cos(x) is "greater than or equal to 1" is exactly the same as saying it is equal to 1. And so the power itself is equal to either 1 or -1. That can only happen when the base is 1 or -1. For what x is cos(x)= 1 or cos(x)= -1?