Question

Prove that : $\cos {20}^o\cos {40}^o\cos {60}^o\cos {80}^o=\frac{1}{16}$

Answer 1

$\cos {20}^{\circ }\cos {40}^{\circ }\cos {60}^{\circ }\cos {80}^{\circ }$

$=\frac{1}{2\sin {20}^{\circ }}\left[2\sin {20}^{\circ }\cos {20}^{\circ }\cos {40}^{\circ }\cos {60}^{\circ }\cos {80}^{\circ }\right]$

$=\frac{1}{2\times 2\sin {20}^{\circ }}\left[2\sin {40}^{\circ }\cos {40}^{\circ }\cos {60}^{\circ }\cos {80}^{\circ }\right]$

$=\frac{1}{2\times 2^2\sin {20}^{\circ }}\left[2\sin {80}^{\circ }\cos {80}^{\circ }\cos {60}^{\circ }\right]$

$=\frac{1}{8\sin {20}^{\circ }}\left[\sin {160}^{\circ }\cos {60}^{\circ }\right]$

$=\frac{1}{8\sin {20}^{\circ }}\left[\sin \left({180}^{\circ }-{20}^{\circ }\right)\times \frac{1}{2}\right]$

$=\frac{1}{8}\times \frac{\sin {20}^{\circ }}{\sin {20}^{\circ }}\times \frac{1}{2}$

$=\frac{1}{16}$