Question

Write the value of the expression
$\frac{1-4sin{10}^{\circ }sin{70}^{\circ }}{2sin{10}^{\circ }}$

Answer 1

We have,
$\frac{1-4sin{10}^{\circ }sin{70}^{\circ }}{2sin{10}^{\circ }}=\frac{1-2\left(2sin{10}^{\circ }sin{70}^{\circ }\right)}{2sin{10}^{\circ }}=\frac{1-2(cos{70}^{\circ }-{10}^{\circ })-cos({70}^{\circ }+{10}^{\circ })}{2sin{10}^{\circ }}=\frac{1-2[cos{60}^{\circ }-cos{80}^{\circ }]}{2sin({90}^{\circ }-{10}^{\circ })}=\frac{1-2(\frac{1}{2}-cos{80}^{\circ })}{2cos{80}^{\circ }}=\frac{1-(\frac{2}{2}+2cos{80}^{\circ })}{2cos{80}^{\circ }}=\frac{1-1+2cos{80}^{\circ }}{2cos{80}^{\circ }}=1$