Problem 218

Analysis

This question examines the trigonometric formulas for the sum and difference of two angles, properties of arithmetic sequences, and the sum of arithmetic sequences. By using the trigonometric formulas for the sum and difference of two angles, we get $\sin^{2}{a_3}\cos^{2}{a_6}-\sin^{2}{a_6}\cos^{2}{a_3}=\sin \left({a_3}-{a_6}\right)\sin \left({a_3}+{a_6}\right)$. Then, using the properties of arithmetic sequences and the formula for the sum of an arithmetic sequence, we find $S_n= -\dfrac{\pi}{12}n^2+\left(a_1+ \dfrac{\pi}{12}\right)n$, and from this, we derive the conclusion.

Solution

Given the arithmetic sequence $\{a_n\}$ satisfies:

$\dfrac{{\sin }^{2}{a_3}-{\cos }^{2}{a_3}+{\cos }^{2}{a_3}\cdot{\cos }^{2}{a_6}-{\sin }^{2}{a_3}\cdot{\sin }^{2}{a_6}}{\sin \left({a_4}+{a_5}\right)}=1$,

$\dfrac{{\sin }^{2}{a_3}(1-{\sin }^{2}{a_6})-{\cos }^{2}{a_3}(1-{\cos }^{2}{a_6})}{\sin \left({a_4}+{a_5}\right)}=1$,

Thus, $\dfrac{{\sin }^{2}{a_3}{\cos }^{2}{a_6}-{\sin }^{2}{a_6}{\cos }^{2}{a_3}}{\sin \left({a_4}+{a_5}\right)}=1$,

Therefore, $\dfrac{\sin \left({a_3}-{a_6}\right)\sin \left({a_3}+{a_6}\right)}{\sin \left({a_3}+{a_6}\right)}=1$,

Hence, $\sin \left({a_3}-{a_6}\right)=1$,

Thus, $a_3-a_6=2k\pi+ \dfrac{\pi}{2}\ (k\in\mathbb{Z})$.

Since $a_3-a_6=-3d \in (0,3)$,

Therefore, $-3d= \dfrac{\pi}{2}$,

Hence, $d=- \dfrac{\pi}{6}$.

Also, since $S_n=n{a_1}+ \dfrac{n(n-1)}{2}d=- \dfrac{\pi}{12}n^2+\left(a_1+ \dfrac{\pi}{12}\right)n$
The equation of the axis of symmetry is $n= \dfrac{6}{\pi}\left(a_1+ \dfrac{\pi}{12}\right)$,
Given that when and only when $n=9$, the sum of the first $n$ terms of the sequence $\{a_n\}$, $S_n$, reaches its maximum value,
Therefore, $\dfrac{17}{2} < \dfrac{6}{\pi}\left(a_1+ \dfrac{\pi}{12}\right) < \dfrac{19}{2}$,

Solving this yields: $\dfrac{4\pi}{3} < a_1 < \dfrac{3\pi}{2}$.

Thus, the correct choice is $\boxed{\text{B}}$.