Period of sine function might be proved as follow.
$$
\begin{array}{rcl}
\sin (t + T) & = & \sin t \newline
\sin t \cos T + \cos t \sin T & = & \newline
\sin t \ (\cos T - 1) & = & - \cos t \sin T \newline
\sin t \cdot 0 & = & - \cos t \cdot 0 \newline
\sin t \ (1 - 1) & = & - \cos t \cdot 0 \newline
\sin t \ (\cos 2\pi - 1) & = & - \cos t \sin 2\pi \newline
T & = & 2\pi
\end{array}
$$
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