GuidedPractice 1. Verify that \( \cot ^{2} \theta-\cos ^{2} \theta=\cot ^{2} \theta \cos ^{2} \theta \) is an identity

To verify that the given expression is an identity, we need to simplify the expression and check if it is equal to zero. Given expression: \( \cot ^{2} \theta - \cos ^{2} \theta = \cot ^{2} \theta \cos ^{2} \theta \) Let's simplify the expression step by step: 1. Simplify the left side of the equation: \( \cot ^{2} \theta - \cos ^{2} \theta \) 2. Simplify the right side of the equation: \( \cot ^{2} \theta \cos ^{2} \theta \) 3. Check if the simplified expressions are equal to each other. Let's simplify the expressions step by step. Simplify the expression by following steps: - step0: Solution: \(\cot^{2}\left(\theta \right)-\cos^{2}\left(\theta \right)\) - step1: Transform the expression: \(\left(\cot\left(\theta \right)+\cos\left(\theta \right)\right)\left(\cot\left(\theta \right)-\cos\left(\theta \right)\right)\) - step2: Transform the expression: \(\frac{\cos\left(\theta \right)+\cos\left(\theta \right)\sin\left(\theta \right)}{\sin\left(\theta \right)}\times \left(\cot\left(\theta \right)-\cos\left(\theta \right)\right)\) - step3: Transform the expression: \(\frac{\cos\left(\theta \right)+\cos\left(\theta \right)\sin\left(\theta \right)}{\sin\left(\theta \right)}\times \frac{\cos\left(\theta \right)-\cos\left(\theta \right)\sin\left(\theta \right)}{\sin\left(\theta \right)}\) - step4: Multiply the terms: \(\frac{\left(\cos\left(\theta \right)+\cos\left(\theta \right)\sin\left(\theta \right)\right)\left(\cos\left(\theta \right)-\cos\left(\theta \right)\sin\left(\theta \right)\right)}{\sin\left(\theta \right)\sin\left(\theta \right)}\) - step5: Multiply the terms: \(\frac{\left(\cos\left(\theta \right)+\cos\left(\theta \right)\sin\left(\theta \right)\right)\left(\cos\left(\theta \right)-\cos\left(\theta \right)\sin\left(\theta \right)\right)}{\sin^{2}\left(\theta \right)}\) - step6: Transform the expression: \(\frac{\cos^{2}\left(\theta \right)\left(1+\sin\left(\theta \right)\right)\left(1-\sin\left(\theta \right)\right)}{\sin^{2}\left(\theta \right)}\) - step7: Transform the expression: \(\frac{\cos^{2}\left(\theta \right)\left(-\left(-\cos^{2}\left(\theta \right)\right)\right)}{\sin^{2}\left(\theta \right)}\) - step8: Calculate: \(\frac{\cos^{2}\left(\theta \right)\cos^{2}\left(\theta \right)}{\sin^{2}\left(\theta \right)}\) - step9: Multiply the terms: \(\frac{\cos^{4}\left(\theta \right)}{\sin^{2}\left(\theta \right)}\) - step10: Transform the expression: \(\cos^{4}\left(\theta \right)\csc^{2}\left(\theta \right)\) - step11: Multiply the terms: \(\csc^{2}\left(\theta \right)\cos^{4}\left(\theta \right)\) Calculate or simplify the expression \( \\cot^{2}(\\theta) \\cos^{2}(\\theta) \). Simplify the expression by following steps: - step0: Solution: \(\cot^{2}\left(\theta \right)\cos^{2}\left(\theta \right)\) - step1: Multiply the terms: \(\left(\cot\left(\theta \right)\cos\left(\theta \right)\right)^{2}\) - step2: Transform the expression: \(\left(\frac{\cos^{2}\left(\theta \right)}{\sin\left(\theta \right)}\right)^{2}\) - step3: Simplify: \(\frac{\cos^{4}\left(\theta \right)}{\sin^{2}\left(\theta \right)}\) - step4: Rewrite the expression: \(\cos^{4}\left(\theta \right)\sin^{-2}\left(\theta \right)\) - step5: Simplify: \(\cos^{4}\left(\theta \right)\left(1+\cot^{2}\left(\theta \right)\right)\) - step6: Rewrite the expression: \(\cos^{4}\left(\theta \right)+\cot^{2}\left(\theta \right)\cos^{4}\left(\theta \right)\) - step7: Calculate: \(\cos^{4}\left(\theta \right)+\left(\csc^{2}\left(\theta \right)-1\right)\cos^{4}\left(\theta \right)\) - step8: Calculate: \(\csc^{2}\left(\theta \right)\cos^{4}\left(\theta \right)\) The simplified expressions are \( \cot ^{2} \theta - \cos ^{2} \theta = \cot ^{2} \theta \cos ^{2} \theta \) and \( \cot ^{2} \theta \cos ^{2} \theta = \cot ^{2} \theta \cos ^{2} \theta \). Therefore, the given expression \( \cot ^{2} \theta - \cos ^{2} \theta = \cot ^{2} \theta \cos ^{2} \theta \) is an identity.