Power Reduction Formulas/Cosine to 5th

Theorem

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

where $\cos$ denotes cosine.

$\ds \cos 5 x$

$=$

$\ds 16 \cos^5 x - 20 \cos^3 x + 5 \cos x$

$\ds \leadsto \ \ $

$\ds 16 \cos^5 x$

$=$

$\ds \cos 5 x + 20 \cos^3 x - 5 \cos x$

rearranging

$\ds $

$=$

$\ds \cos 5 x + 20 \paren {\frac {3 \cos x + \cos 3 x} 4} - 5 \cos x$

$\ds $

$=$

$\ds \cos 5 x + 15 \cos x + 5 \cos 3 x - 5 \cos x$

multipying out

$\ds $

$=$

$\ds 10 \cos x + 5 \cos 3 x + \cos 5 x$

rearranging

$\ds \leadsto \ \ $

$\ds \cos^5 x$

$=$

$\ds \frac {10 \cos x + 5 \cos 3 x + \cos 5 x} {16}$

dividing both sides by 16

$\blacksquare$