UTAR Question

$\sin20\sin40\sin60\sin80$

$=\sin20\sin(60-20)\sin60\sin(60+20)$

$=\sin20[\sin60\cos20-\cos60\sin20]\sin60[\sin60\cos20+\cos60\sin20]$

$=(\sin20)(\frac{\sqrt3}{2})(\frac{\sqrt3}{2}\cos20-\frac{1}{2}\sin20)(\frac{\sqrt3}{2}\cos20+\frac{1}{2}\sin20)$

$=\frac{\sqrt3}{8}\sin20(\sqrt{3}\cos20-\sin20)(\sqrt{3}\cos20+\sin20)$

$=\frac{\sqrt3}{8}\sin20(3\cos^2 20-\sin^2 20)$

$=\frac{\sqrt3}{8}\sin20(3(1-\sin^2 20)-\sin^2 20)$

$=\frac{\sqrt3}{8}\sin20(3-3\sin^2 20-\sin^2 20)$

$=\frac{\sqrt3}{8}\sin20(3-4\sin^2 20)$

$=\frac{\sqrt3}{8}(3\sin20-4\sin^3 20)$

$=\frac{\sqrt3}{8}(\sin(3 \times 20))$

$=\frac{\sqrt3}{8}(\sin60)$

$=(\frac{\sqrt3}{8})(\frac{\sqrt3}{2})=\frac{3}{16}$