Period of trig functions

Question

How to prove the periodicity and then find the primitive period of summation, products and compositions of trig functions? Is it possible to prove that the primitive period of sine function is $2\pi$ using its series definition? How to prove that the primitive period of tangent function is $\pi$ knowing that $\tan x =\frac{\sin x}{\cos x}$ and that the primitive period of sine and cosine is $2\pi$? Thanks in advance.