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

Answer: 2.00

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