<s>[INST] Evaluating $\frac {\cos 81^{\circ}}{\sin3^{\circ}\sin57^{\circ}\sin63^{\circ}} $ 
Find $$\frac {\cos 81^{\circ}}{\sin3^{\circ}\sin57^{\circ}\sin63^{\circ}} $$

This expression seemed quite easy to solve, but now the $63$ and $57$ in the equation are posing me with difficulties. I tried multiplying by $\cos9$ but I was unable to solve it further. 
Would someone please help me to solve this question? 
Thanks in advance. 
 [/INST] Let $\alpha = 60^\circ$ and $\theta = 3^\circ$. The numerator is
$$\begin{align}
\cos(81^\circ) = \sin(9^\circ) &= \sin(3\theta)\\
&= \sin\theta\cos(2\theta) + \cos\theta\sin(2\theta)\\
&= \sin\theta(\cos(2\theta) + 2\cos^2\theta)\\
&= \sin\theta(2\cos(2\theta) + 1)\end{align}$$
while the denominator equals to
$$\begin{align}
\sin 3^\circ\sin 57^\circ \sin 63^\circ
&= \sin\theta\sin(\alpha-\theta)\sin(\alpha+\theta)\\
&= \frac12\sin\theta(\cos(2\theta) - \cos(2\alpha))\\
&= \frac14\sin\theta(2\cos(2\theta) + 1)\end{align}$$
This means the expression at hand equals to $4$.
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