Power Reduction Formulas/Sine to 5th

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Theorem

$\sin^5 x = \dfrac {10 \sin x - 5 \sin 3 x + \sin 5 x} {16}$

where $\sin$ denotes sine.


Proof

\(\ds \sin 5 x\) \(=\) \(\ds 5 \sin x - 20 \sin^3 x + 16 \sin^5 x\) Quintuple Angle Formula for Sine
\(\ds \leadsto \ \ \) \(\ds 16 \sin^5 x\) \(=\) \(\ds \sin 5 x + 20 \sin^3 x - 5 \sin x\) rearranging
\(\ds \) \(=\) \(\ds \sin 5 x + 20 \paren {\frac {3 \sin x - \sin 3 x} 4} - 5 \sin x\) Power Reduction Formula for Cube of Sine
\(\ds \) \(=\) \(\ds \sin 5 x + 15 \sin x - 5 \sin 3 x - 5 \sin x\) multipying out
\(\ds \) \(=\) \(\ds 10 \sin x - 5 \sin 3 x + \sin 5 x\) rearranging
\(\ds \leadsto \ \ \) \(\ds \sin^5 x\) \(=\) \(\ds \frac {10 \sin x - 5 \sin 3 x + \sin 5 x} {16}\) dividing both sides by 16

$\blacksquare$


Sources