Question

Find the value of $\frac{\cos 20^{\circ}+8 \sin 70^{\circ} \sin 50^{\circ} \sin 10^{\circ}}{\sin ^{2} 80^{\circ}}$.

Answer 1

$\frac{\cos 20+8 \sin 70 \cdot \sin 50 \sin 10}{\sin ^{2} 80}$
$=\frac{\cos 20+8 \sin 10 \sin (60-10) \sin (60+10)}{\sin ^{2} 80}$
$\{\because \sin A \sin (60-A) \sin (60+A) =\sin 3 A / 4$
$=\frac{\cos 20+\frac{8 \sin 30}{4}}{\sin ^{2} 80}$
$=\frac{1+\cos 20}{\frac{1-\cos 160}{2}}$
$=\frac{2(1+\cos 20)}{1+\cos 20}=2$