Prove that \( \sin x+\sin 3 x+\ldots+\sin (2 n-1) x=\frac{\sin ^{2} n x}{\sin x} \) for all \( n...

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Prove that \( \sin x+\sin 3 x+\ldots+\sin (2 n-1) x=\frac{\sin ^{2} n x}{\sin x} \) for all \( n \in N \).
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