Question

Evaluate $\frac{\sin 9^{\circ }\times \cos 9^{\circ }}{\sin {48}^{\circ }\times \cos {12}^{\circ }}$

Answer 1

Given $\frac{\sin 9^{\circ }\times \cos 9^{\circ }}{\sin {48}^{\circ }\times \cos {12}^{\circ }}$
Dividing and multiplying by $2$, we get
$=\frac{2\sin 9^{\circ }\times \cos 9^{\circ }}{2\sin {48}^{\circ }\times \cos {12}^{\circ }}$
Use, formula, $2\sin A.\cos A=\sin 2A$
and $2\sin A.\cos B=\sin (A+B)+\sin (A-B)$
$=\frac{\sin {18}^{\circ }}{\sin ({48}^{\circ }+{12}^{\circ })+sin({48}^{\circ }-{12}^{\circ })}$
$=\frac{\sin {18}^{\circ }}{\sin {60}^{\circ }+\sin {36}^{\circ }}$
we know,
$\sin {18}^{\circ }=\frac{\sqrt{5}-1}{4}$
$\sin {36}^{\circ }=\frac{\sqrt{10-2\sqrt{5}}}{4}$
Therefore, the value of given expression is $\frac{\frac{\sqrt{5}-1}{4}}{\frac{\sqrt{3}}{2}+\frac{\sqrt{10-2\sqrt{5}}}{4}}$
$=\frac{\frac{\sqrt{5}-1}{4}}{\frac{2\sqrt{3}+\sqrt{10-2\sqrt{5}}}{4}}$
$=\frac{\sqrt{5}-1}{2\sqrt{3}+\sqrt{10-2\sqrt{5}}}$