Power Reduction Formulas/Cosine to 5th

(Redirected from Fifth Power of Cosine)

Theorem

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

where $\cos$ denotes cosine.

Proof

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

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