Prove the identity. \[ \tan \theta(\tan \theta+\cot \theta)=\sec ^{2} \theta \] To verify the identity, work each side separately until you obtain the same expression. Start with the left side. Choose the correct step and transform the expression according to the step chosen. \[ \tan \theta(\tan \theta+\cot \theta)=\square \] Distribute.

Let's start by taking the left side of the identity: \[ \tan \theta(\tan \theta+\cot \theta) \] Now we can distribute \(\tan \theta\): \[ \tan \theta \cdot \tan \theta + \tan \theta \cdot \cot \theta \] This simplifies to: \[ \tan^2 \theta + 1 \] Since we know from the Pythagorean identity that \(\sec^2 \theta = 1 + \tan^2 \theta\), we can further simplify: \[ \tan^2 \theta + 1 = \sec^2 \theta \] Thus, we have shown that: \[ \tan \theta(\tan \theta+\cot \theta) = \sec^2 \theta \] So the identity is proved to be correct!