Step 1: Rewrite in degrees:
Convert to degrees to avoid fractions of $\pi$. The principal values are $\tan^{-1}\sqrt{3}=60^{\circ}$ and $\sin^{-1}(-\frac12)=-30^{\circ}$.
Step 2: Handle the secant term:
Use $\sec^{-1}(-x)=\pi-\sec^{-1}x$ for $x\geq1$. Here $\sec^{-1}2=\frac{\pi}{3}$, because $\cos\frac{\pi}{3}=\frac12$. So $\sec^{-1}(-2)=\pi-\frac{\pi}{3}=120^{\circ}$.
Step 3: Combine:
The expression is $60^{\circ}+120^{\circ}-(-30^{\circ})=210^{\circ}$. Since $180^{\circ}=\pi$, we get $210^{\circ}=\frac{210}{180}\pi=\frac{7\pi}{6}$.
Step 4: Check the options:
$\frac{5\pi}{6}=150^{\circ}$, $\frac{2\pi}{3}=120^{\circ}$ and $\frac{4\pi}{3}=240^{\circ}$. Only $\frac{7\pi}{6}=210^{\circ}$ matches our total.
Final Answer:
The value is $210^{\circ}$, which equals $\frac{7\pi}{6}$ (option C).
\[ \boxed{\dfrac{7\pi}{6}} \]