$$\begin{aligned} & \cos 2 x+a \cdot \sin x=2 a-7 \\ & a(\sin x-2)=2(\sin x-2)(\sin x+2) \\ & \sin x=2, a=2(\sin x+2) \\ & \Rightarrow a \in[2,6] \\ & p=2 \quad q=6 \\ & r=\tan 9^{\circ}+\cot 9^{\circ}-\tan 27-\cot 27 \\ & r=\frac{1}{\sin 9 \cdot \cos 9}-\frac{1}{\sin 27 \cdot \cos 27} \\ & =2\left[\frac{4}{\sqrt{5}-1}-\frac{4}{\sqrt{5}+1}\right] \\ & r=4 \\ & p \cdot q \cdot r=2 \times 6 \times 4=48 \end{aligned}$$