Vorily the identity \[ \cos (\alpha+\beta) \cos (\alpha-\beta)=\cos ^{2} \alpha-\sin ^{2} \beta \] Write the left side of the identity using the sum and difference formula. ( \( \cos \alpha \cos \beta-\sin \alpha \sin \beta)(\cos \alpha \cos \beta+\sin \alpha \sin \beta) \) (Type the terms of your expression in the same order as they appear in the original expression.) Use the distributive property to simplify the expression from the previous step. \( \cos { }^{2} \alpha \cos ^{2} \beta-\sin ^{2} \alpha \sin ^{2} \beta \) (Simplify your answer.) Substitute \( 1-\sin ^{2} \beta \) for \( \cos ^{2} \beta \) in the expression from the previous step and distribute. \( \square \)

To prove the identity, we start by substituting the terms using the sum and difference formulas for cosine: \[ (\cos \alpha \cos \beta - \sin \alpha \sin \beta)(\cos \alpha \cos \beta + \sin \alpha \sin \beta) \] When we apply the distributive property (also known as the FOIL method), we have: \[ \cos^2 \alpha \cos^2 \beta - \sin^2 \alpha \sin^2 \beta \] Now, simplifying this further leads us to substitute \( \cos^2 \beta \) with \( 1 - \sin^2 \beta \): \[ \cos^2 \alpha (1 - \sin^2 \beta) - \sin^2 \alpha \sin^2 \beta \] Distributing \( \cos^2 \alpha \) across the expression gives us: \[ \cos^2 \alpha - \cos^2 \alpha \sin^2 \beta - \sin^2 \alpha \sin^2 \beta \] This gives us the expression we need to work with for further simplification or to reach the final identity.