Use \( \cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta \) to expand \( \cos 2 \alpha \) 6

To expand \( \cos 2\alpha \), we can use the angle addition formula. Set \( \beta = \alpha \), and we get: \[ \cos 2\alpha = \cos(\alpha + \alpha) = \cos \alpha \cos \alpha - \sin \alpha \sin \alpha = \cos^2 \alpha - \sin^2 \alpha. \] This gives us the double angle formula for cosine: \( \cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha \). Now, if you're feeling adventurous, remember that you can rewrite \( \cos^2 \alpha \) as \( 1 - \sin^2 \alpha \) or vice versa! This can lead to another popular form of the double angle identity: \( \cos 2\alpha = 1 - 2\sin^2 \alpha \) or \( \cos 2\alpha = 2\cos^2 \alpha - 1 \). Play around with it and see which you prefer!