Your calculations are almost all good, except that you make a mistake in the last 2 lines. It is actually

This helps by getting that

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So now, the sum on top is

[tex]\displaystyle \sin 3\theta+\sin 5\theta+\sin 7\theta+... = \frac{1}{2\sin\theta}\left(\sum_{r = 1}^n (f(r)-f(r+1)\right)[/tex]

Which is a telescoping sum which sums to

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