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.

Calculate or simplify the expression \( \tan(\theta) \cdot (\tan(\theta) + \cot(\theta)) \). Simplify the expression by following steps: - step0: Solution: \(\tan\left(\theta \right)\left(\tan\left(\theta \right)+\cot\left(\theta \right)\right)\) - step1: Transform the expression: \(\frac{\sin\left(\theta \right)}{\cos\left(\theta \right)}\times \left(\tan\left(\theta \right)+\cot\left(\theta \right)\right)\) - step2: Transform the expression: \(\frac{\sin\left(\theta \right)}{\cos\left(\theta \right)}\times \frac{\sin^{2}\left(\theta \right)+\cos^{2}\left(\theta \right)}{\cos\left(\theta \right)\sin\left(\theta \right)}\) - step3: Reduce the fraction: \(\frac{1}{\cos\left(\theta \right)}\times \frac{\sin^{2}\left(\theta \right)+\cos^{2}\left(\theta \right)}{\cos\left(\theta \right)}\) - step4: Multiply the terms: \(\frac{\sin^{2}\left(\theta \right)+\cos^{2}\left(\theta \right)}{\cos\left(\theta \right)\cos\left(\theta \right)}\) - step5: Multiply the terms: \(\frac{\sin^{2}\left(\theta \right)+\cos^{2}\left(\theta \right)}{\cos^{2}\left(\theta \right)}\) - step6: Rewrite the expression: \(\frac{1}{\cos^{2}\left(\theta \right)}\) - step7: Transform the expression: \(1+\tan^{2}\left(\theta \right)\) - step8: Transform the expression: \(1+\sec^{2}\left(\theta \right)-1\) - step9: Remove the opposites: \(\sec^{2}\left(\theta \right)\) The left side of the identity is \(\tan \theta(\tan \theta+\cot \theta)=\sec ^{2} \theta\). Now, let's work on the right side of the identity. The right side of the identity is \(\sec ^{2} \theta\). Therefore, the identity \(\tan \theta(\tan \theta+\cot \theta)=\sec ^{2} \theta\) is verified.