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