2.12 Use 2.11 to write down two other expressions representing \( \cos 2 \alpha \) HINT USE THE Identity \( \sin ^{2} \theta+\cos ^{2} \theta=1 \)

Let's have some fun with trigonometric identities! If we start from the identity given in 2.11, \( \cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha \), we can also express this in two other forms. First, using the Pythagorean identity where \( \sin^2 \alpha = 1 - \cos^2 \alpha \), we can substitute to get \( \cos 2\alpha = \cos^2 \alpha - (1 - \cos^2 \alpha) = 2\cos^2 \alpha - 1 \). Now, let’s flip it! By substituting \( \cos^2 \alpha = 1 - \sin^2 \alpha \), we can rewrite \( \cos 2\alpha = (1 - \sin^2 \alpha) - \sin^2 \alpha = 1 - 2\sin^2 \alpha \). So there you have them: \( \cos 2\alpha = 2\cos^2 \alpha - 1 \) and \( \cos 2\alpha = 1 - 2\sin^2 \alpha \)! Trig identities can be quite the playful puzzle!