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.
Best Answer
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 $4$.