Evaluate Principal Value of Composite Inverse Trigonometric Expression

To evaluate the given expression: $E = \cot^{-1}(\cot(-11)) + 10 \sin\left(2 \cos^{-1}\left(\frac{1}{\sqrt{2}}\right)\right) + 10 \sin(2 \tan^{-1}(2))$

We evaluate each term individually using the principal values of the inverse trigonometric functions:

1. For the first term, since $4\pi - 11 \in (0, \pi)$ and $\cot(4\pi - 11) = \cot(-11)$, we have: $\cot^{-1}(\cot(-11)) = 4\pi - 11$

2. For the second term, using $\cos^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4}$: $10 \sin\left(2 \cdot \frac{\pi}{4}\right) = 10 \sin\left(\frac{\pi}{2}\right) = 10$

3. For the third term, using the identity $\sin(2\tan^{-1} x) = \frac{2x}{1+x^2}$ with $x=2$: $10 \sin(2 \tan^{-1}(2)) = 10 \cdot \frac{2(2)}{1 + 2^2} = 10 \cdot \frac{4}{5} = 8$

Summing the results from all three terms yields: $E = (4\pi - 11) + 10 + 8 = 4\pi + 7$

Thus, the correct option is (C).