How to show that a function is periodic?

Question

I have this function $y(t)=6\sin{t}-6+\frac{a}{e^{\sin(t)}}+\frac{6}{e^{\sin(t)}} , a\in{\mathbb{R}}$ where i'm trying to show that its periodic.

I have attempted to show that $y(t)=y(t+P)$ , here i can use the fact that $\sin{t}$ is periodic but i'm having trouble showing the composition $e^{\sin{t}}$ is periodic.

Answer 1

Composition preserves periodicity. That is, if $f(t)$ is periodic with period $T$, and $h = g(f(t))$, then $h$ should be periodic with the same period. More precisely: $$h(t+T) = g(f(t+T)) = g(f(t)) = h(t) $$