a) Find the value of $m$ if $\sec 65^{\circ}=\operatorname{cosec}(2 m-15)^{\circ}$

Step 1 ：Given that \(\sec 65^\circ = \operatorname{cosec}(2m - 15)^\circ\)

Step 2 ：Using the identities \(\sec x = \frac{1}{\cos x}\) and \(\operatorname{cosec} x = \frac{1}{\sin x}\), we get \(\frac{1}{\cos 65^\circ} = \frac{1}{\sin (2m - 15)^\circ}\)

Step 3 ：Using the identity \(\sin (2x) = 2 \sin x \cos x\), we can rewrite the equation as \(\frac{1}{\cos 65^\circ} = \frac{1}{2 \sin (m - 7.5^\circ) \cos (m - 7.5^\circ)}\)

Step 4 ：Solving for m, we get \(m = 20\)

Step 5 ：\(\boxed{m = 20}\)