1. XERCISE 2 1. Prove the following: (a) \( \cos (\alpha+\beta)+\cos (\alpha-\beta)=2 \cos \alpha, \cos \beta \) (b) \( \sin \left(\theta+30^{\circ}\right)-\sin \left(\theta-30^{\circ}\right)=\cos \theta \) (c) \( \cos \left(45^{\circ}-\theta\right)-\sin \left(45^{\circ}-\theta\right)=\sqrt{2} \sin \theta \)

To prove the given trigonometric identities, we will use known trigonometric identities and properties. Let's go through each part step by step. ### (a) Prove that \( \cos (\alpha+\beta) + \cos (\alpha-\beta) = 2 \cos \alpha \cos \beta \) **Step 1: Use the cosine addition and subtraction formulas.** The cosine addition and subtraction formulas are: \[ \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \] \[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \] **Step 2: Add the two equations.** Now, we add these two equations: \[ \cos(\alpha + \beta) + \cos(\alpha - \beta) = (\cos \alpha \cos \beta - \sin \alpha \sin \beta) + (\cos \alpha \cos \beta + \sin \alpha \sin \beta) \] **Step 3: Simplify the expression.** This simplifies to: \[ \cos(\alpha + \beta) + \cos(\alpha - \beta) = 2 \cos \alpha \cos \beta \] Thus, we have proved that: \[ \cos (\alpha+\beta) + \cos (\alpha-\beta) = 2 \cos \alpha \cos \beta \] ### (b) Prove that \( \sin \left(\theta + 30^{\circ}\right) - \sin \left(\theta - 30^{\circ}\right) = \cos \theta \) **Step 1: Use the sine subtraction formula.** The sine subtraction formula is: \[ \sin(A) - \sin(B) = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \] **Step 2: Apply it to our case.** Let \( A = \theta + 30^{\circ} \) and \( B = \theta - 30^{\circ} \): \[ \sin\left(\theta + 30^{\circ}\right) - \sin\left(\theta - 30^{\circ}\right) = 2 \cos\left(\frac{(\theta + 30^{\circ}) + (\theta - 30^{\circ})}{2}\right) \sin\left(\frac{(\theta + 30^{\circ}) - (\theta - 30^{\circ})}{2}\right) \] **Step 3: Simplify the arguments.** Calculating the averages: \[ \frac{(\theta + 30^{\circ}) + (\theta - 30^{\circ})}{2} = \frac{2\theta}{2} = \theta \] \[ \frac{(\theta + 30^{\circ}) - (\theta - 30^{\circ})}{2} = \frac{60^{\circ}}{2} = 30^{\circ} \] **Step 4: Substitute back into the equation.** Thus, we have: \[ \sin\left(\theta + 30^{\circ}\right) - \sin\left(\theta - 30^{\circ}\right) = 2 \cos(\theta) \sin(30^{\circ}) \] Since \( \sin(30^{\circ}) = \frac{1}{2} \): \[ = 2 \cos(\theta) \cdot \frac{1}{2} = \cos(\theta) \] Thus, we have proved that: \[ \sin \left(\theta + 30^{\circ}\right) - \sin \left(\theta - 30^{\circ}\right) = \cos \theta \] ### (c) Prove that \( \cos \left(45^{\circ} - \theta\right) - \sin \left(45^{\circ} - \theta\right) = \sqrt{2} \sin \theta \) **Step 1: Use the cosine and sine of a difference.** Using the cosine and sine of a difference: \[ \cos(45^{\circ} - \theta) = \cos 45^{\circ} \cos \theta + \sin 45^{\circ} \sin \theta \] \[ \sin(45^{\circ} - \theta) = \sin 45^{\circ} \cos \theta - \cos 45^{\circ} \sin \theta \] **Step 2: Substitute the values of \( \cos 45^{\circ} \) and \( \sin 45^{\circ} \).** Since \( \cos 45^{\circ} = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \): \[ \cos(45^{\circ} - \theta) = \frac{\sqrt{2}}{2} \cos \theta + \frac{\sqrt{2}}{2} \sin \theta \] \[ \sin(45^{\circ} - \theta) = \frac{\sqrt{2}}{2} \cos \theta - \frac{\sqrt{2}}{2} \sin \theta \] **Step 3: Substitute into the equation.** Now, substituting these into the left-hand side: \[ \cos(45^{\circ} - \theta) - \sin(45^{\circ} - \theta) = \left(\frac{\sqrt{2}}{2} \cos \theta + \frac{\sqrt{2}}{2} \sin \theta\right) - \left(\frac{\sqrt{2}}{2} \cos \theta - \frac{\sqrt{2}}{2} \sin \theta\right) \] **Step 4: Simplify the expression.** This simplifies to: \[ = \frac{\sqrt{2}}{2} \cos \theta + \frac{\sqrt{2}}{2} \sin \theta - \frac{\sqrt{2}}{2} \cos \theta + \frac{\sqrt{2}}{2} \sin \theta \] \[ = \sqrt{2} \sin \theta \] Thus, we have proved that: \[ \cos \left(45^{\circ} - \theta\right) - \sin \left(45^{\circ} - \theta\right) = \sqrt{2} \sin \theta \] ### Conclusion We have successfully proved all three identities: 1. \( \cos (\alpha+\beta) + \cos (\alpha-\beta) = 2 \cos \alpha \cos \beta \) 2. \( \sin \left(\theta + 30^{\circ}\right) - \sin \left(\theta - 30^{\