4 Sine Pi over 10 by Cosine Pi over 5/Proof 3

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Theorem

$4 \sin \dfrac \pi {10} \cos \dfrac \pi 5 = 1$


Proof

\(\ds 4 \sin \theta \cos 2 \theta\) \(=\) \(\ds 1\) Solve for $\theta$
\(\ds 4 \sin \theta \cos \theta \cos 2\theta\) \(=\) \(\ds \cos \theta\) multiplying both sides by $\cos \theta$
\(\ds 2 \paren {2 \sin \theta \cos \theta } \cos 2\theta\) \(=\) \(\ds \cos \theta\) factoring out $2$
\(\ds 2 \paren {\sin 2 \theta } \cos 2\theta\) \(=\) \(\ds \cos \theta\) Double Angle Formula for Sine
\(\ds \sin 4 \theta\) \(=\) \(\ds \cos \theta\) Double Angle Formula for Sine
\(\ds \map \sin {\frac \pi 2 - \theta}\) \(=\) \(\ds \cos \theta\) Sine of Complement equals Cosine
\(\ds \paren {\frac \pi 2 - \theta}\) \(=\) \(\ds 4 \theta\)
\(\ds \theta\) \(=\) \(\ds \frac \pi {10}\)

$\blacksquare$