What is the period of $(2007)^{\sin x}$?

$f(x+2\pi) = 2007^{\sin(x+2\pi)} = 2007^{\sin(x)} = f(x)$

Period of $(2007)^{\sin x}$ = period of $\sin x$ = $2\pi$ since the periodicity of the function depends on $\sin x$ only, irrespective that is in the exponent.

$$\psi(x+2\pi)=(2007)^{\sin (x+2\pi)}=(2007)^{\sin x}=\psi(x)$$