While studying trigonometric functions I keep encountering exercises of this sort:
$f\left(x\right)=\frac{\tan\left(x\right)}{1+\sin\left(x\right)}$. Show that the periodicity of $f(x)$ is $2\pi $.
I have looked on the Internet and the most I could find was stating that since the periodicity of $\textbf{tan} =\pi$ and the periodicity of $\textbf{sin}=2\pi$, then the compound periodicity is $2\pi$. But why?
I'm learning Math all by myself, so I really can't afford to just memorize algorithms and formulas. Could you provide me with a thorough explanation of the process by which I can arrive at such conclusions? I would mostly value and algebraic proof.