\section{Power Reduction Formulas/Cosine to 5th}
Tags: Cosine Function

\begin{theorem}
:$\cos^5 x = \dfrac {10 \cos x + 5 \cos 3 x + \cos 5 x} {16}$
\end{theorem}

\begin{proof}
{{begin-eqn}}
{{eqn | l = \cos 5 x
      | r = 16 \cos^5 x - 20 \cos^3 x + 5 \cos x
      | c = [[Quintuple Angle Formula for Cosine]]
}}
{{eqn | ll= \leadsto
      | l = 16 \cos^5 x
      | r = \cos 5 x + 20 \cos^3 x - 5 \cos x
      | c = rearranging
}}
{{eqn | r = \cos 5 x + 20 \paren {\frac {3 \cos x + \cos 3 x} 4} - 5 \cos x
      | c = [[Power Reduction Formula for Cube of Sine]]
}}
{{eqn | r = \cos 5 x + 15 \cos x + 5 \cos 3 x - 5 \cos x
      | c = multipying out
}}
{{eqn | r = 10 \cos x + 5 \cos 3 x + \cos 5 x
      | c = rearranging
}}
{{eqn | ll= \leadsto
      | l = \cos^5 x
      | r = \frac {10 \cos x + 5 \cos 3 x + \cos 5 x} {16}
      | c = dividing both sides by 16
}}
{{end-eqn}}
{{qed}}
\end{proof}


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