Sum of the series $\sinθ\sin2θ + \sin2θ\sin3θ + \sin3θ\sin4θ + \sin4θ\sin5θ + \cdots+\sin n\theta\sin(n+1)\theta$ terms The series is given:
$$\sum_{i=1}^n \sin (i\theta) \sin ((i+1)\theta)$$
 We have to find the sum to n terms of the given series.
 I could took out the $2\sin^2\cos$ terms common in the series. But what to do further, please guide me.

Hint: For the part $A = \displaystyle \sum_{i=1}^{n} \cos (2i+1)x$, use:
$\sin x\displaystyle \sum_{i=1}^n \cos (2i+1)x$, and expand each term you can see a telescope sum.
