91Ó°ÊÓ

First, express \(\cos 2\theta\) as \(1 - 2\sin^2\theta\). This transforms the left hand side of the equation to \(\frac{1 - 2\sin^2\theta}{\cos\theta}\). To further simplify, express \(\sin^2\theta\) and \(\cos^2\theta\) in terms of \(\tan\theta\) and \(\sec\theta\). Solving the \(\tan\theta\) identity for \(\sin^2\theta\) gives \(\sin^2\theta = \tan^2\theta\cos^2\theta\). Substitute the expression for \(\sin^2\theta\) back into the equation from the first step to get: \(\frac{1 - 2(\tan^2\theta\cos^2\theta)}{\cos\theta}\). Further replace \(\cos^2\theta\) with \(\frac{1}{\sec^2\theta}\) leading to \(\frac{1 - 2(\frac{\tan^2\theta}{\sec^2\theta})}{\cos\theta}\). Finally, simplifying the equation leads to \(\frac{\sec^2\theta - 2\tan^2\theta}{\sec\theta}\) which further simplifies to \(\frac{1-\tan^2\theta}{\sec\theta}\) proving the given identity.