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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 ;)

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    $\begingroup$ Can you use de moive theorem and taking the imaginary part? ? $\endgroup$
    – Chinny84
    Dec 2, 2014 at 16:51
  • $\begingroup$ I would get a common denominator and work with trig identities on the numerator. I'm not sure if that will work, but it seems a reasonable place to start. $\endgroup$
    – KSmarts
    Dec 2, 2014 at 16:54

2 Answers 2

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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 $1$.

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  • $\begingroup$ This is just beautiful. Thank you ;) $\endgroup$ Dec 2, 2014 at 17:24
  • $\begingroup$ It is satisfying to see a very bare bones idea I had being fleshed out by you :)! Well done +1 $\endgroup$
    – Chinny84
    Dec 2, 2014 at 17:34
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Hint:

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

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