The question text appears to have a formatting issue. Interpreting it in the standard way, the expression is:

$$\left(\sin \frac{11x}{2}\right)(\sin 6x-\cos 6x)+\left(\cos \frac{11x}{2}\right)(\sin 6x+\cos 6x)$$

and we need to simplify it.

1. Expand and regroup

Let

$$E=\sin \frac{11x}{2}(\sin 6x-\cos 6x)+\cos \frac{11x}{2}(\sin 6x+\cos 6x)$$

Expanding,

$$E=\sin \frac{11x}{2}\sin 6x-\sin \frac{11x}{2}\cos 6x+\cos \frac{11x}{2}\sin 6x+\cos \frac{11x}{2}\cos 6x$$

Group the first and last pair suitably:

$$E=\left(\sin \frac{11x}{2}\sin 6x+\cos \frac{11x}{2}\cos 6x\right) +\left(-\sin \frac{11x}{2}\cos 6x+\cos \frac{11x}{2}\sin 6x\right)$$

2. Use trigonometric identities

We use:

$$\cos(A-B)=\cos A\cos B+\sin A\sin B$$

and

$$\sin(B-A)=\sin B\cos A-\cos B\sin A$$

So,

$$\sin \frac{11x}{2}\sin 6x+\cos \frac{11x}{2}\cos 6x =\cos\left(6x-\frac{11x}{2}\right)$$

and

$$-\sin \frac{11x}{2}\cos 6x+\cos \frac{11x}{2}\sin 6x =\sin\left(6x-\frac{11x}{2}\right)$$

Hence,

$$E=\cos\left(6x-\frac{11x}{2}\right)+\sin\left(6x-\frac{11x}{2}\right)$$

Now,

$$6x-\frac{11x}{2}=\frac{12x-11x}{2}=\frac{x}{2}$$

Therefore,

$$E=\sin\frac{x}{2}+\cos\frac{x}{2}$$

3. Compare with options

The simplified expression is

$$\boxed{\sin\frac{x}{2}+\cos\frac{x}{2}}$$

But all the given options are constants:

• A: $\dfrac{\sqrt{11}-1}{2\sqrt{3}}$
• B: $\dfrac{\sqrt{11}+1}{2\sqrt{3}}$
• C: $\dfrac{\sqrt{11}+1}{3\sqrt{2}}$
• D: $\dfrac{\sqrt{11}-1}{3\sqrt{2}}$

So the expression as written cannot equal one fixed constant for all $x$.

This indicates the question is incomplete or misprinted. Possibly some condition on $x$ is missing.

4. Conclusion

From the given expression,

$$\boxed{\sin\frac{x}{2}+\cos\frac{x}{2}}$$

is the correct simplification.

So none of the listed options matches the expression as currently written.