calculus - trigonometric identity related to $ sum_{n=1}^{infty}frac{sin(n)sin(n^{2})}{sqrt{n}} $

As homework I was given the following series to check for convergence:

$ \displaystyle \sum_{n=1}^{\infty}\dfrac{\sin(n)\sin(n^{2})}{\sqrt{n}} $

and the tip was "use the appropriate identity".

I'm trying to use Dirichlet's test and show that it's the product of a null monotonic sequence and a bounded series, but I can't figure out which trig. identity is needed.

Can anyone point me towards the right direction?

Many thanks.

Answer

Hint: You can show that $$ \sum\limits_{n=1}^N\sin(n)\sin(n^2)=\frac{1}{2}(1-\cos(N^2+N)) $$ To do this use identity $$ \sin(\alpha)\sin(\beta)=\frac{1}{2}(\cos(\alpha-\beta)-\cos(\alpha+\beta)) $$