Explanation
Given cos2x+asinx=2a−7
⇒1−2sin2x+7=−asin2x+2a
⇒2(4−sin2x)=a(2−sinx)
\begin{aligned}
& \Rightarrow2\left(2+\sin x\right)=a \\
& \Rightarrow a=4+2\sin x-1\leq\sin x\leq1 \\
& \Rightarrow-2\leq2\sin\leq2 \\
& \Rightarrow2\leq a\leq6 \\
& \therefore p=2 & q=6 \\
& r=\tan9^{\circ}-\tan27^{\circ}-\frac{1}{\cos63^{\circ}}+\tan81^{\circ} \\
& =\tan9^{\circ}+\cot9^{\circ}-\tan27^{\circ}-\cot27^{\circ} \\
& =\frac{\sin9^{\circ}}{\cos9^{\circ}}+\frac{\cos9^{\circ}}{\sin^{\circ}9}-\left[\frac{\sin27^{\circ}}{\cos27^{\circ}}+\frac{\cos27^{\circ}}{\sin27^{\circ}}\right] \\
& =\frac{\sin^{2}9^{\circ}+\cos^{2}9^{\circ}}{\sin9^{\circ}\mathrm{cos}9^{\circ}}-\left[\frac{\sin^{2}27^{\circ}+\cos^{2}27^{\circ}}{\sin27^{\circ}\mathrm{cos}27^{\circ}}\right] \\
& =\frac{2}{\sin18^{\circ}}-\frac{2}{\sin54^{\circ}} \\
& =2\left\{\frac{4}{\sqrt{5}-1}-\frac{4}{\sqrt{5}+1}\right\}=8\left\{\frac{\sqrt{5}+1}{4}-\frac{\sqrt{5}-1}{4}\right\} \\
& =2\{2\}=4 \\
& \therefore pqr=2\times6\times4=48
\end{aligned}
