How do i find this $k$?

Question

I don't know where my teacher got this monster from . I'd appriciate if you could help me with this.

Question: Let $$f_n(x)=\sum^n_{r=1}\frac{\sin^2x}{\cos^2 \frac{x}{2}-\cos^2\frac{(2r+1)x}{2}}$$ And $$g_n(x)=\prod^n_{r=1}f_r(x) $$

and$$T_n=\int^\pi_0\frac{f_n(x)}{g_n(x)}$$

Then $$\sum^{10}_{r=1}T_r=k\pi$$

What is the value of $k$?

Attempt: I have not gone far yet. I can see that $\cos^2A-\cos^2B$ ought to be some trigonometric identity like $\sin (A+B).sin(A-B)$ Therefore my $f_n$ boils down to this:

$$\sum^n_{r=1}\frac{\sin^2x}{\sin (nx) * \sin (n+1)x}$$

But how to proceed further?A bit detailing will be appreciated. Thanks

Answer 1

HINT:

$$\sin x=\sin\{(n+1)x-\sin nx\}=?$$

Can you recognize the Telescoping series?