Calculate the sum : $S=1+\frac{\sin x}{\sin x}+\frac{\sin2x}{\sin^2 x}+\cdots +\frac{\sin nx}{\sin^n x}$

Question

I was given a task to calculate this sum: $$S=1+\frac{\sin x}{\sin x}+\frac{\sin 2x}{\sin^2 x}+\cdots +\frac{\sin nx}{\sin^n x}$$ but I'm not really sure how to start solving it. Like always, I would be grateful if someone could provide a subtle hint on how to begin. Thanks ;)

Answer 1

Define $S^{'}=S-1$

Let $C^{'}=\dfrac{\cos x}{\sin x} + \dfrac{\cos2x}{\sin^2x} + ....... + \dfrac{\cos nx}{\sin^nx}$

Now,

$S^{'}=\Im{(C^{'}+iS^{'})}=\dfrac{e^{ix}}{\sin x} + \dfrac{e^{2ix}}{\sin^2x} + ...... + \dfrac{e^{nix}}{\sin^nx}$

Now sum it as a G.P. and get the imaginary part of the sum and lastly, add $1$.

Answer 2

Hint:

$\dfrac{\sin ((n+1)x)}{\sin^{n+1} x}=\dfrac{\sin (nx + x)}{\sin^n x \times \sin x}= \dfrac{\sin nx \times \cos x + \sin x \times \cos nx}{\sin^n x \times \sin x}$