Answer:
[tex]\frac{sin\alpha.cos\beta +cos\alpha.sin\beta}{cos\alpha.cos\beta}=tan\alpha +tan\beta[/tex]
Explanation:
Here are the steps to simplify the expression and obtain the identity.
1) Given:
[tex]\frac{si(\alpha+\beta)}{cos(\alpha)cos(\beta)}[/tex]
2) Use the identity sin(α + β) = sin(α)cos(β) + cos(α)sin(β)
[tex]\frac{sin\alpha.cos\beta +cos\alpha.sin\beta}{cos\alpha.cos \beta}[/tex]
3) Distributive property of division:
[tex]\frac{sin\alpha.cos\beta}{cos\alpha.cos \beta}+\frac{cos\alpha.sin\beta}{cos\alpha.cos \beta}[/tex]
4) Simplify common factors in numerators and denominators
[tex]\frac{sin\alpha}{cos\alpha }+\frac{sin\beta}{cos \beta}[/tex]
5) Definition of tangent ratio
tanα + tanβ
6) Conclusion
[tex]\frac{sin\alpha.cos\beta +cos\alpha.sin\beta}{cos\alpha.cos \beta}=tan\alpha +tan\beta[/tex]
