Simplify: \(\frac{1+\tan^2(x)}{1+\cot^2(x)}\times\frac{\sec(x)-\tan(x)}{\sec(x)+\tan(x)}\)

Solution

Using \(1+\tan^2x=\sec^2x\) and \(1+\cot^2x=\csc^2x\), the first fraction simplifies to \(\frac{\sec^2x}{\csc^2x}=\tan^2x\). Multiplying the numerator and denominator of the second fraction by \(\cos x\) gives \(\frac{1-\sin x}{1+\sin x}\). Therefore, the expression becomes \(\frac{\tan^2x(1-\sin x)}{1+\sin x}\). Trick: Apply the Pythagorean identities first, then multiply the second fraction by \(\frac{\cos x}{\cos x}\).