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

$4 \sin \dfrac \pi {10} \cos \dfrac \pi 5 = 1$
 $\ds 4 \sin \theta \cos 2 \theta$ $=$ $\ds 1$ 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 $\ds \paren {\frac \pi 2 - \theta}$ $=$ $\ds 4 \theta$ $\ds \theta$ $=$ $\ds \frac \pi {10}$
$\blacksquare$