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.

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.

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## Comments

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TopNewestIt 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 · 1 month, 1 week ago

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