Question

Simplify:

$\sin 36.\sin 27.\sin 108.\sin 144=\frac{5}{16}$.

Answer 1

$\sin 36°\sin 72°\sin 108°\sin 144°$

$=\sin 36°\sin 72°\sin (180°-72°)\sin (180°-36°)$

$=\sin 36°\sin 72°\sin (90°\ast 2)-72°\sin (90°\ast 2)-36°$

$=\sin 36°\sin 72°\sin 72°\sin 36°$

$={\sin }^{2}36°{\sin }^{2}72°$

$={\left[\frac{\sqrt{10}-2\sqrt{5}}{4}\right]}^2\ast {\left[\frac{\sqrt{10}+2\sqrt{5}}{4}\right]}^2$

$=\left[\frac{\sqrt{10}-2\sqrt{5}}{16}\right]\ast \left[\frac{\sqrt{10}+2\sqrt{5}}{16}\right]$

$=\frac{(100-20)}{(16\times 16)}$

$=\frac{80}{(16\times 16)}$

$=\frac{5}{16}$(Proved)