cos 2 x + a ⋅ sin x = 2 a − 7 a ( sin x − 2 ) = 2 ( sin x − 2 ) ( sin x + 2 ) sin x = 2 , a = 2 ( sin x + 2 ) ⇒ a ∈ [ 2 , 6 ] p = 2 q = 6 r = tan 9 ∘ + cot 9 ∘ − tan 27 − cot 27 r = 1 sin 9 ⋅ cos 9 − 1 sin 27 ⋅ cos 27 = 2 [ 4 5 − 1 − 4 5 + 1 ] r = 4 p ⋅ q ⋅ r = 2 × 6 × 4 = 48 \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} cos 2 x + a ⋅ sin x = 2 a − 7 a ( sin x − 2 ) = 2 ( sin x − 2 ) ( sin x + 2 ) sin x = 2 , a = 2 ( sin x + 2 ) ⇒ a ∈ [ 2 , 6 ] p = 2 q = 6 r = tan 9 ∘ + cot 9 ∘ − tan 27 − cot 27 r = sin 9 ⋅ cos 9 1 − sin 27 ⋅ cos 27 1 = 2 [ 5 − 1 4 − 5 + 1 4 ] r = 4 p ⋅ q ⋅ r = 2 × 6 × 4 = 48