We have $\sin^{2n}(x)+\cos^{2n}(x)$ has period $\pi/2$ for $n\in\mathbb{N}\backslash \{1\}$:

Also $\sin^{2n+1}(x)+\cos^{2n+1}(x)$ has period $2\pi$ for $n\in\mathbb{N}$:

We have $\sin^{-2n}(x)+\cos^{-2n}(x)$ has period $\pi/2$ for $n\in\mathbb{N}$:

Also $\sin^{-2n+1}(x)+\cos^{-2n+1}(x)$ has period $2\pi$ for $n\in\mathbb{N}$: