In a triangle \(\mathrm{ABC}\) with usual notations \(\mathrm{a}=2, \mathrm{~b}=3\), then value of \(\frac{\cos 2 A}{a^2}-\frac{\cos 2 B}{b^2}\) is

\(\frac{\cos 2 \mathrm{~A}}{\mathrm{a}^2}-\frac{\cos 2 \mathrm{~B}}{\mathrm{~b}^2} \)
\( =\frac{1-2 \sin ^2 \mathrm{~A}}{\mathrm{a}^2}-\frac{1-2 \sin ^2 \mathrm{~B}}{\mathrm{~b}^2}=\left(\frac{1}{\mathrm{a}^2}-\frac{1}{\mathrm{~b}^2}\right)-2\) \(\left(\frac{\sin ^2 \mathrm{~A}}{\mathrm{a}^2}-\frac{\sin ^2 \mathrm{~B}}{\mathrm{~b}^2}\right) \)
From sine rule, we know that
\(\frac{\mathrm{a}}{\sin \mathrm{A}}=\frac{\mathrm{b}}{\sin \mathrm{B}} \)
\( \therefore \text { Given Expression }=\left(\frac{1}{\mathrm{a}^2}-\frac{1}{\mathrm{~b}^2}\right)-0=\frac{1}{2^2}-\frac{1}{3^2}\) \(=\frac{1}{4}-\frac{1}{9}=\frac{5}{36}\)