Simplify: \(15 + \dfrac{12}{\cot^2 A} - \dfrac{12}{\cos^2 A}\)
3
Rewrite each fraction using reciprocal identities. Since \(\frac{1}{\cot^2 A} = \tan^2 A\) and \(\frac{1}{\cos^2 A} = \sec^2 A\), the expression becomes \(15 + 12\tan^2 A - 12\sec^2 A\).
Factor the trigonometric terms: \(15 + 12(\tan^2 A - \sec^2 A)\).
Use the Pythagorean identity \(\sec^2 A - \tan^2 A = 1\), which gives \(\tan^2 A - \sec^2 A = -1\).
Substitute: \(15 + 12(-1) = 15 - 12 = 3\).
Hence, the simplified value is 3.
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