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

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

$\blacksquare$


Sources