Express $\cot A-\tan A$ entirely in terms of $\sin A$ and $\cos A$.

$\cot A-\tan A=\frac{\cos A}{\sin A}-\frac{\sin A}{\cos A}=\frac{{\cos }^{2}A-{\sin }^{2}A}{\sin A\cos A}$. The option with ${\sin }^{2}A-{\cos }^{2}A$ on top is the same fraction with the sign reversed — the order of subtraction must be preserved. Note that it is $\cot A+\tan A$, not the difference, that collapses to $\frac{1}{\sin A\cos A}$.