Step 1: Write the ratios in terms of sine and cosine instead of quoting cot and sec directly.
Recall that $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$ and $\sec\theta = \dfrac{1}{\cos\theta}$.
Step 2: Work out $\cot^{2}30^{\circ}$ from sine and cosine.
We know $\sin 30^{\circ} = \dfrac{1}{2}$ and $\cos 30^{\circ} = \dfrac{\sqrt{3}}{2}$, so:
\[ \cot^{2}30^{\circ} = \frac{\cos^{2}30^{\circ}}{\sin^{2}30^{\circ}} = \frac{\left(\frac{\sqrt{3}}{2}\right)^{2}}{\left(\frac{1}{2}\right)^{2}} = \frac{\frac{3}{4}}{\frac{1}{4}} = 3 \]
Step 3: Work out $\sec^{2}60^{\circ}$ from cosine.
We know $\cos 60^{\circ} = \dfrac{1}{2}$, so:
\[ \sec^{2}60^{\circ} = \frac{1}{\cos^{2}60^{\circ}} = \frac{1}{\left(\frac{1}{2}\right)^{2}} = \frac{1}{\frac{1}{4}} = 4 \]
Step 4: Substitute into the given expression.
\[ \frac{1}{3}(3) - \frac{1}{2}(4) = 1 - 2 = -1 \]
Final Answer:
The value of the expression is $-1$, matching option (A).
\[ \boxed{-1} \]