Step 1: Principal values
Evaluate each term numerically in degrees: $\sin 150^\circ = 0.5$, so $\sin^{-1}(0.5) = 30^\circ$.
Step 2: Next terms
$\cos 210^\circ = -\frac{\sqrt{3}}{2}$, and $\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) = 150^\circ$. $\tan 120^\circ = -\sqrt{3}$, so $\tan^{-1}(-\sqrt{3}) = -60^\circ$.
Step 3: Add
$30^\circ + 150^\circ - 60^\circ = 120^\circ = \frac{2\pi}{3}$.
Step 4: Result
Option (A). The values $\frac{4\pi}{3}$, $\frac{5\pi}{3}$ and $\pi$ correspond to $240^\circ$, $300^\circ$ and $180^\circ$, which do not match.
Final Answer:
The sum is 120 degrees, i.e. 2 pi / 3. This is option (A).
\[ \boxed{\text{(A) }\frac{2\pi}{3}} \]