Step 1: Plan:
Convert everything to degrees, then take the ratio.
Step 2: Degrees:
Numerator: $45^{\circ} + 60^{\circ} + 30^{\circ} = 135^{\circ}$. Denominator: $\tan^{-1}\sqrt3 = 60^{\circ}$, and $\sec^{-1}(-2)$ is the angle in $[0,\pi]$ with secant $-2$, i.e. cosine $-\frac12$, which is $120^{\circ}$. So denominator $= 60^{\circ} - 120^{\circ} = -60^{\circ}$.
Ratio $= \frac{135}{-60} = -\frac{9}{4}$.
Step 3: Common slip:
If one takes $\sec^{-1}(-2) = -\frac{\pi}{3}$, the denominator becomes $\frac{2\pi}{3}$ and the ratio comes out as $\frac{9}{8}$, which is wrong. The range of $\sec^{-1}$ is $[0,\pi]$ except $\frac{\pi}{2}$, so a negative argument gives an obtuse angle.
Final Answer:
The value is $-\frac{9}{4}$, option (D).
\[ \boxed{-\frac{9}{4}} \]