If \( x \) is not an integral multiple of \( \pi \), use induction ...

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If \( x \) is not an integral multiple of \( \pi \), use induction to prove that
\[
\sin x+\sin 3 x+\sin 5 x+\ldots+\sin (2 n-1) x=\frac{\sin ^{2} n x}{\sin x} \text { for all } n \in \mathbf{N} .
\]
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