\( 2 \sin ^{2} \theta \cos ^{2} \theta+\cos ^{4} \theta=1-\sin ^{4} \theta \) 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. \[ 2 \sin ^{2} \theta \cos ^{2} \theta+\cos ^{4} \theta=\square \]

To verify the identity \( 2 \sin ^{2} \theta \cos ^{2} \theta+\cos ^{4} \theta=1-\sin ^{4} \theta \), we will start with the left side and simplify it step by step. Starting with the left side: \[ 2 \sin^{2} \theta \cos^{2} \theta + \cos^{4} \theta \] ### Step 1: Factor out \(\cos^{2} \theta\) We can factor out \(\cos^{2} \theta\) from the first term: \[ = \cos^{2} \theta (2 \sin^{2} \theta + \cos^{2} \theta) \] ### Step 2: Use the Pythagorean identity We know that \(\sin^{2} \theta + \cos^{2} \theta = 1\). Therefore, we can express \(2 \sin^{2} \theta + \cos^{2} \theta\) as follows: \[ = \cos^{2} \theta (2 \sin^{2} \theta + (1 - \sin^{2} \theta)) \] This simplifies to: \[ = \cos^{2} \theta (2 \sin^{2} \theta + 1 - \sin^{2} \theta) \] ### Step 3: Combine like terms Now, combine the terms inside the parentheses: \[ = \cos^{2} \theta (1 + \sin^{2} \theta) \] ### Step 4: Substitute back Now we can express this as: \[ = \cos^{2} \theta + \sin^{2} \theta \cos^{2} \theta \] ### Step 5: Use the Pythagorean identity again Using the identity \(\sin^{2} \theta + \cos^{2} \theta = 1\): \[ = \cos^{2} \theta + \sin^{2} \theta \cos^{2} \theta \] This is the left side simplified. Now we can express it as: \[ = \cos^{2} \theta (1 + \sin^{2} \theta) \] Now we can proceed to simplify the right side \(1 - \sin^{4} \theta\) to see if it matches. ### Right Side: Starting with the right side: \[ 1 - \sin^{4} \theta \] ### Step 1: Factor the expression We can factor \(1 - \sin^{4} \theta\) as a difference of squares: \[ = (1 - \sin^{2} \theta)(1 + \sin^{2} \theta) \] ### Step 2: Use the Pythagorean identity Using \(1 - \sin^{2} \theta = \cos^{2} \theta\): \[ = \cos^{2} \theta (1 + \sin^{2} \theta) \] ### Conclusion Now we have both sides simplified to: \[ \cos^{2} \theta (1 + \sin^{2} \theta) \] Thus, we have verified that: \[ 2 \sin^{2} \theta \cos^{2} \theta + \cos^{4} \theta = 1 - \sin^{4} \theta \] The identity holds true.