(Solved): Derive the identity for \( \tan (\alpha+\beta) \) using \( \tan (\alpha+\beta)=\frac{\sin (\alpha+ ...

Derive the identity for \( \tan (\alpha+\beta) \) using \( \tan (\alpha+\beta)=\frac{\sin (\alpha+\beta)}{\cos (\alpha+\beta)} \). After applying the formulas for sums of sines and cosines, divide the numerator and denominator by \( \cos \alpha \cos \beta \). Apply the formulas for sums of sines and cosines. \[ \tan (\alpha+\beta)=\frac{\sin (\alpha+\beta)}{\cos (\alpha+\beta)} \] Divide the numerator and denominator by \( \cos \alpha \cos \beta \). \[ \tan (\alpha+\beta)=\frac{\overline{\cos \alpha \cos \beta}}{\frac{\cos \alpha \cos \beta}{}} \] Divide each term in the numerator and denominator by \( \cos \alpha \cos \beta \). \[ \tan (\alpha+\beta)=\frac{\frac{\sin \alpha \cdot}{\cos \alpha \cos \beta}+\frac{\cos \alpha \cdot}{\cos \alpha \cos \beta}}{\frac{\cos \alpha \cdot}{\cos \alpha \cos \beta}-\frac{\sin \alpha \cdot}{\cos \alpha \cos \beta}} \] cosines, divide the numerator and denominator by \( \cos \alpha \cos \beta \). \( =\square \) umerator and denominator by \( \cos \alpha \cos \beta \). Divide each term in the numerator and denominator by \( \cos \alpha \cos \beta \). \( \tan (\alpha+\beta)=\frac{\frac{\sin \alpha \cdot}{\cos \alpha \cos \beta}+\frac{\cos \alpha \cdot}{\cos \alpha \cos \beta}}{\frac{\cos \alpha \cdot}{\cos \alpha \cos \beta}-\frac{\sin \alpha \cdot}{\cos \alpha \cos \beta}} \) \( =\frac{1+1 \cdot}{1 \cdot 1-\cdot \tan \beta} \) \( = \) \[ =\frac{\square}{\square} \] Divide the numerator and denominator by \( \cos \alpha \cos \beta \). \[ \tan (\alpha+\beta)=\frac{\overline{\cos \alpha \cos \beta}}{\frac{}{\cos \alpha \cos \beta}} \] Divide each term in the numerator and \( d \) \( \tan (\alpha+\beta)=\frac{\frac{\sin \alpha \cdot}{\cos \alpha \cos \beta}+\frac{\cos \alpha \cdot}{\cos \alpha \cos }}{\frac{\cos \alpha \cdot}{\cos \alpha \cos \beta}-\frac{\sin \alpha \cdot}{\cos \alpha \operatorname{co}}} \quad \) Use the re \( =\frac{\cdot 1+1 \cdot}{1 \cdot 1-\cdot \tan \beta} \) \( = \)