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Sine Omega Identity

Prove that

\[4 \sin (x) \sin (\omega x) \sin (\omega^2 x) = - (\sin (2x) + \sin (2 \omega x) + \sin (2 \omega^2 x))\]

\[\] Notation: \(\omega\) denotes a primitive cube root of unity.

Note by Ishan Singh
3 months, 4 weeks ago

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It is easy because it can be done in one line (brute force) if we express both sides in exponential form (Euler's formula). I find it interesting because there are some beautiful generalizations and elegant proofs for the same. We can also use this identity to express them as infinite sums using partial fraction. Ishan Singh · 3 months, 4 weeks ago

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