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.

Hint1:
$$\sum_{i=1}^n \sin (i\theta) \sin ((i+1)\theta)
=  1/2\sum_{i=1}^n \left(\cos (\theta)-\cos ((2i+1)\theta)\right)$$
Hint2:
$$ 2\sin (\theta)\sum_{i=1}^n \cos ((2i+1)\theta) = \sum_{i=1}^n \left(\sin ((2i+2)\theta) - \sin (2i\theta)\right) $$
