Find the Sum

$\small \begin{aligned} S & = f(h) + f \left( h + \frac \pi 2 \right) + f(h + \pi) + f \left( h + \frac {3\pi} 2 \right) + \cdots + f(h + 99\pi) + f \left(h + \frac {199\pi} 2 \right) \\ & = \blue{\sin (2h)} + \red{\sin (\pi + 2h)} + \blue{\sin (2\pi + 2h)} + \red{\sin (3\pi + 2h)} + \cdots + \blue{\sin (198\pi + 2h)} + \red{\sin (199\pi + 2h)} \\ & = \blue{\sin (2h)} + \red{\sin (\pi + 2h)} + \blue{\sin (2h)} + \red{\sin (\pi + 2h)} + \blue{\sin (2h)} + \red{\sin (\pi + 2h)} + \cdots + \blue{\sin (2h)} + \red{\sin (\pi + 2h)} \\ & = \blue{\sin (2h)} \red{- \sin (2h)} \blue{+ \sin (2h)} \red{- \sin (2h)}\blue{+ \sin (2h)} \red{- \sin (2h)}\blue{+ \sin (2h)} \red{- \sin (2h)}+\cdots \blue{+ \sin (2h)} \red{- \sin (2h)} \\ & = \boxed 0 \end{aligned}$

$\begin{aligned} & f\left( h \right)+f\left( h+\frac{\pi }{2} \right)+f\left( h+\pi \right)+f\left( h+\frac{3\pi }{2} \right)+f\left( h+2\pi \right)+\ldots +f\left( h+99\pi \right)+f\left( h+\frac{199\pi }{2} \right) \\ & =\sum\limits_{k=0}^{199}{f\left( h+\frac{k\pi }{2} \right)} \\ & =\sum\limits_{k=0}^{199}{\sin \left( 2h+k\pi \right)} \\ & =\sum\limits_{n=0}^{99}{\sin \left( 2h+2n\pi \right)}+\sum\limits_{n=0}^{99}{\sin \left[ 2h+\left( 2n+1 \right)\pi \right]} \\ & =\sum\limits_{n=0}^{99}{\sin \left( 2h \right)}+\sum\limits_{n=0}^{99}{\sin \left[ 2h+\pi \right]} \\ & =\sum\limits_{n=0}^{99}{\sin \left( 2h \right)}+\sum\limits_{n=0}^{99}{\left[ -\sin \left( 2h \right) \right]} \\ & =\sum\limits_{n=0}^{99}{\sin \left( 2h \right)}-\sum\limits_{n=0}^{99}{\sin \left( 2h \right)} \\ & =\boxed{0} \\ \end{aligned}$