Considering only the principal values of the inverse trigonometric functions, evaluate:

${\cot }^{-1}\left(\cot \right(-11\left)\right)+10\sin \left(\mathrm{(2}{\cos }^{-(}\right)$ ¹ ( 1 /2 ))+ 10 sin 2 tan ^{− 1} ( 2 )

`. Let's double-check all MathML in the solution to ensure no HTML entities for minus, and no `display="block"`. Everything is clean and straightforward. Let's write out the response. $4\pi +7$

The correct option is $4\pi +7$.

To evaluate the given expression using the principal values of the inverse trigonometric functions, we break the expression into three separate terms:

$E={\cot }^{-1}\left(\cot \right(-11\left)\right)+10\sin \left(2{\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)\right)+10\sin \left(2{\tan }^{-1}\left(2\right)\right)$

Step 1: Evaluate the first term, $T_1={\cot }^{-1}\left(\cot \right(-11\left)\right)$

Recall that the principal value branch of ${\cot }^{-1}\left(x\right)$ is $(0,\pi )$.

We know that cotangent is periodic with period $\pi$, so $\cot (-11)=\cot (4\pi -11)$.

Since $\pi \approx 3.14159$, we have $4\pi \approx 12.56637$.

Therefore, $4\pi -11\approx 1.56637$, which lies in the principal interval $(0,\pi )$.

Hence:

$T_1={\cot }^{-1}\left(\cot \right(4\pi -11\left)\right)=4\pi -11$

Step 2: Evaluate the second term, $T_2=10\sin \left(2{\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)\right)$

The principal value of ${\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)$ is $\frac{\pi }{4}$.

Substituting this value in:

$T_2=10\sin \left(2\times \frac{\pi }{4}\right)=10\sin \left(\frac{\pi }{2}\right)$

Since $\sin \left(\frac{\pi }{2}\right)=1$:

$T_2=10\times 1=10$

Step 3: Evaluate the third term, $T_3=10\sin \left(2{\tan }^{-1}\left(2\right)\right)$

Let $\theta ={\tan }^{-1}\left(2\right)$, so that $\tan \left(\theta \right)=2$.

Using the double-angle identity for sine in terms of tangent:

$\sin \left(2\theta \right)=\frac{2\tan \left(\theta \right)}{1+{\tan }^2\left(\theta \right)}=\frac{2\left(2\right)}{1+2^2}=\frac{4}{5}$

Therefore:

$T_3=10\times \frac{4}{5}=8$

Step 4: Combine all terms

Adding the results of the three terms together:

$E=T_1+T_2+T_3=(4\pi -11)+10+8$

$E=4\pi -11+18=4\pi +7$