Step 1: Convert each term to a known angle:
Use the odd and complementary identities of inverse functions.
$\tan^{-1}(-x) = -\tan^{-1}x$ and $\cos^{-1}(-x) = \pi - \cos^{-1}x$.
Step 2: Use the standard angle table:
$\tan\frac{\pi}{4}=1$, $\cos\frac{\pi}{3}=\frac12$ and $\sin\frac{\pi}{6}=\frac12$.
So $\tan^{-1}(-1)=-\frac{\pi}{4}$, $\cos^{-1}\left(-\frac12\right)=\pi-\frac{\pi}{3}$ and $\sin^{-1}\frac12=\frac{\pi}{6}$.
Step 3: Add in degrees to avoid fractions:
In degrees the terms are $-45^\circ$, $120^\circ$ and $30^\circ$.
Their sum is $-45+120+30=105$ degrees.
Since $180^\circ=\pi$, we get $105^\circ=\frac{105\pi}{180}=\frac{7\pi}{12}$.
Step 4: Test the options numerically:
The sum is about $-0.7854+2.0944+0.5236=1.8326$.
Option (A) $\frac{7\pi}{12}\approx1.8326$ matches.
Option (B) is about $1.309$, option (C) is negative and option (D) is about $1.047$, so none of them fits.
Final Answer:
The correct choice is option (A).
\[ \boxed{\frac{7\pi}{12}} \]