Cosine of 72 Degrees/Proof 2

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Theorem

$\cos 72 \degrees = \cos \dfrac {2 \pi} 5 = \dfrac {\sqrt 5 - 1} 4$


Proof

\(\ds \cos 72 \degrees\) \(=\) \(\ds 2 \cos 36 \degrees - 1\)
\(\ds \) \(=\) \(\ds 2 \paren {\dfrac \phi 2}^2 - 1\) Cosine of $36 \degrees$
\(\ds \) \(=\) \(\ds \dfrac {\phi^2} 2 - 1\)
\(\ds \) \(=\) \(\ds \dfrac {\phi + 1} 2 - 1\) Square of Golden Mean equals One plus Golden Mean
\(\ds \) \(=\) \(\ds \dfrac {\phi - 1} 2\)
\(\ds \) \(=\) \(\ds -\dfrac {1 - \phi} 2\)
\(\ds \) \(=\) \(\ds \dfrac {\phi^{-1} } 2\) Reciprocal Form of One Minus Golden Mean
\(\ds \) \(=\) \(\ds \dfrac {\sqrt 5 - 1} 4\) Definition 2 of Golden Mean, and algebra

$\blacksquare$


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