<s>[INST] What is $\sin x \sin 2x \sin 4x$ converted to a sum? Converting production to sum. I tried to find the formulas but i could find the formulas only for two identities not for three that is why I couldn't solve. Anybody help?
 [/INST] $$ \begin{align*} \sin x \sin 2x \sin 4x &= \frac{1}{2}(\cos x - \cos 3x) \sin 4x \\&= \frac{1}{2}\cos x \sin 4x - \frac{1}{2}\cos 3x \sin 4x \\&= \frac{1}{4}(\sin 5x + \sin 3x) - \frac{1}{4}(\sin 7x + \sin x) \\&= \frac{1}{4} (\sin 5x + \sin 3x - \sin 7x - \sin x) \end{align*}$$
</s>