Power Reduction Formulas/Cosine to 7th

Theorem

$\cos^7 x = \dfrac {35 \cos x + 21 \cos 3 x + 7 \cos 5 x + \cos 7 x} {64}$

where $\cos$ denotes cosine.

$\ds \cos 7 x$

$=$

$\ds 64 \cos^7 x - 112 \cos^5 x + 56 \cos^3 x - 7 \cos x$

$\ds \leadsto \ \ $

$\ds 64 \cos^7 x$

$=$

$\ds \cos 7 x + 112 \cos^5 x - 56 \cos^3 x + 7 \cos x$

rearranging

$\ds $

$=$

$\ds \cos 7 x + 112 \paren {\dfrac {10 \cos x + 5 \cos 3 x + \cos 5 x} {16} } - 56 \paren {\frac {3 \cos x + \cos 3 x} 4} + 7 \cos x$

$\ds $

$=$

$\ds \cos 7 x + 70 \cos x + 35 \cos 3 x + 7 \cos 5 x - 42 \cos x - 14 \cos 3 x + 7 \cos x$

multipying out

$\ds $

$=$

$\ds 35 \cos x + 21 \cos 3 x + 7 \cos 5 x + \cos 7 x$

rearranging

$\ds \leadsto \ \ $

$\ds \cos^7 x$

$=$

$\ds \frac {35 \cos x + 21 \cos 3 x + 7 \cos 5 x + \cos 7 x} {64}$

dividing both sides by $64$

$\blacksquare$