1.4 Use \( 213[\cos (\alpha-\beta)=\cos \alpha \cdot \cos \beta+\sin \alpha \cdot \sin \beta] \) to derive a formuls for \( \cos \{\alpha+\beta \) (Hint: use suitable reduction formula)

1. Start with the given identity: \[ \cos(\alpha - \beta) = \cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta. \] 2. Replace \(\beta\) with \(-\beta\). This gives: \[ \cos(\alpha - (-\beta)) = \cos \alpha \cdot \cos(-\beta) + \sin \alpha \cdot \sin(-\beta). \] 3. Simplify the left-hand side: \[ \cos(\alpha + \beta) = \cos \alpha \cdot \cos(-\beta) + \sin \alpha \cdot \sin(-\beta). \] 4. Use the even-odd properties of the cosine and sine functions: - Cosine is even: \(\cos(-\beta) = \cos \beta\). - Sine is odd: \(\sin(-\beta) = -\sin \beta\). 5. Substitute these into the equation: \[ \cos(\alpha + \beta) = \cos \alpha \cdot \cos \beta + \sin \alpha \cdot (-\sin \beta). \] 6. Simplify the expression: \[ \cos(\alpha + \beta) = \cos \alpha \cdot \cos \beta - \sin \alpha \cdot \sin \beta. \]