j) \( \frac{\cot A \cos A}{\sec ^{2} A-1} \)

Did you know that the expression you provided can be simplified? The denominator, \( \sec^2 A - 1 \), is equivalent to \( \tan^2 A \) based on the Pythagorean identity. So your expression simplifies to \( \frac{\cot A \cos A}{\tan^2 A} \). And remember that \( \cot A = \frac{\cos A}{\sin A} \), so when you substitute that in, it leads you to unravel the wonders of trigonometric identities! If you’re grappling with trigonometric simplifications, it’s always a good idea to work with the fundamental identities. For instance, transforming all expressions into sine and cosine can often clarify the path to simplification. Just remember, common mistakes include overlooking the Pythagorean identities or misapplying angle sum formulas. Wield those identities like a wizard’s wand, and watch the complexities vanish!