2.12 Use 2.11 to write down two other expressions representing \( \cos 2 \alpha \) HINT: USE THE identity \( \sin ^{2} \theta+\cos ^{2} \theta=1 \) 2.13 Use 2.9 to expand \( \sin 2 \alpha \)

We begin with the identity given by equation 2.11 for the double angle for cosine: \[ \cos 2\alpha = \cos^2\alpha - \sin^2\alpha. \] **Step 1. Expressing \(\cos 2\alpha\) in two other ways using \(\sin^2\theta+\cos^2\theta=1\):** 1. Replace \(\sin^2\alpha\) using the identity: \[ \sin^2\alpha = 1 - \cos^2\alpha. \] Substituting this into the expression for \(\cos 2\alpha\) gives: \[ \cos 2\alpha = \cos^2\alpha - (1 - \cos^2\alpha) = 2\cos^2\alpha - 1. \] 2. Alternatively, replace \(\cos^2\alpha\) using the identity: \[ \cos^2\alpha = 1 - \sin^2\alpha. \] Substituting this into the expression for \(\cos 2\alpha\) gives: \[ \cos 2\alpha = (1 - \sin^2\alpha) - \sin^2\alpha = 1 - 2\sin^2\alpha. \] Thus, we have two additional expressions: \[ \cos 2\alpha = 2\cos^2\alpha - 1, \quad \text{and} \quad \cos 2\alpha = 1 - 2\sin^2\alpha. \] **Step 2. Expanding \(\sin 2\alpha\) using the identity from 2.9:** The double angle formula for sine is: \[ \sin 2\alpha = 2\sin\alpha\cos\alpha. \] Thus, the expanded form is: \[ \sin 2\alpha = 2\sin\alpha\cos\alpha. \]