Sine Party!

$\begin{aligned} S & = \frac{\sin 1^\circ}{\sin 45^\circ \sin 46^\circ} + \frac{\sin 1^\circ}{\sin 47^\circ \sin 48^\circ} + \frac{\sin 1^\circ}{\sin 49^\circ \sin 50^\circ} + ... + \frac{\sin 1^\circ}{\sin 133^\circ \sin 134^\circ} \\ & = \sum_{n=0}^{44} \frac{\sin 1^\circ}{\sin (45+2n)^\circ \sin (45 +2n+1)^\circ} \\ & = \sum_{n=0}^{44} \frac{\sin (45+2n+1 - 45+2n)^\circ}{\sin (45+2n)^\circ \sin (45 +2n+1)^\circ} \\ & = \sum_{n=0}^{44} \frac{\sin (45+2n+1)^\circ \cos (45+2n)^\circ - \sin (45+2n)^\circ \cos (45+2n+1)^\circ}{\sin (45+2n)^\circ \sin (45 +2n+1)^\circ} \\ & = \sum_{n=0}^{44} (\cot (45+2n)^\circ - \cot (45+2n+1)^\circ) \\ & = \sum_{n=45}^{134} (-1)^{n+1}\cot n^\circ \\ & = \sum_{n=45}^{89} (-1)^{n+1}\cot n^\circ - \cot 90^\circ + \color{#3D99F6}{\sum_{n=91}^{134} (-1)^{n+1}\cot n^\circ} \quad \quad \small \color{#3D99F6}{\text{As }\cot (180^\circ-\theta) = - \cot \theta} \\ & = \sum_{n=45}^{89} (-1)^{n+1}\cot n^\circ - 0 + \color{#3D99F6}{\sum_{n=91}^{134} (-1)^{n}\cot (180-n)^\circ} \\ & = \sum_{n=45}^{89} (-1)^{n+1}\cot n^\circ + \color{#3D99F6}{\sum_{n=46}^{89} (-1)^{n}\cot n^\circ} \\ & = \cot 45^\circ \\ & = \boxed{1} \end{aligned}$