Let the set of all $a\in R$ such that the equation $\cos 2x+a\sin x=2a-7$ has a solution be  $\left[p,q\right]$ and $r=\tan 9^0-\tan {27}^0-\frac{1}{\cot {63}^0}+\tan {81}^0$ then pqr is equal to 

Given $\cos 2x+a\sin x=2a-7$
$1-2{\sin }^{2}x+a\sin x=2a-7$
$2{\sin }^{2}x-a\sin x+2a-8=0put\sin x=t$
$\begin{array}{rl}2t^2-at+2a-8& =0\\ (2t^2-8)-a(t-2)& =0\\ 2(t^2-4)-a(t-2)& =0\\ 2(t+2)(t-2)-a(t-2)& =0\\ (t-2)(2t+4-a)& =0\end{array}$
$t=2\Rightarrow t=\frac{a-4}{2}$
$t=2\Rightarrow \sin x=2\notin \left[-1,1\right]\Rightarrow \sin x=\frac{a-4}{2}$
$\begin{array}{c}-1\le \frac{a-4}{2}\le 1\\ -2\le a-4\le 2\\ 2\le a\le 6\end{array}$
$\begin{array}{c}a\in [2,6]\Rightarrow [p,q]\\ p=2,q=6\end{array}$
$\begin{array}{rl}r& =\tan 9^{\circ }-\tan {27}^{\circ }-\frac{1}{\cot {63}^{\circ }}+\tan {81}^{\circ }\\ & =\tan 9^{\circ }-\tan {27}^{\circ }-\tan {63}^{\circ }+\tan {81}^{\circ }\\ & =\tan 9^{\circ }+\cot 9^{\circ }-(\tan {27}^{\circ }+\cot {27}^{\circ })\end{array}$
$=2\cos ec{18}^0-2\cos ec{54}^0=2\left[\frac{4}{\sqrt{5}-1}-\frac{4}{\sqrt{5}+1}\right]$
$=2\times 4\left[\frac{\sqrt{5}+1-\sqrt{5}+1}{4}\right]$
$=2\times 2=4$
$pqr=2\times 6\times 4=48$