<s>[INST] Find sum of $\frac{1}{\sin\theta\cdot \sin2\theta} + \frac{1}{\sin2\theta\cdot \sin3\theta} + \cdots + \frac{1}{\sin n \theta \sin (n+1)\theta}$ $$\sum_{k=1}^n \frac{1}{\sin k\theta \sin (k+1)\theta} = \dfrac{1}{\sin\theta\cdot \sin2\theta} + \dfrac{1}{\sin2\theta\cdot \sin3\theta} + \cdots + \frac{1}{\sin n \theta \sin (n+1)\theta}$$ up to $n$ terms.
I tried but in vain
 [/INST] Here is a hint: the sum telescopes, with some careful manipulation:
$$\sin \theta = \sin((k+1)\theta - k\theta) = \sin(k+1)\theta \cos k\theta - \sin k\theta \cos (k+1)\theta.$$
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