Question

Complete the identity. (sin(\alpha +\beta ))/(cos\alpha cos\beta )

Answer 1

Answer:

$\frac{sin\alpha.cos\beta +cos\alpha.sin\beta}{cos\alpha.cos\beta}=tan\alpha +tan\beta$

Explanation:

Here are the steps to simplify the expression and obtain the identity.

1) Given:

$\frac{si(\alpha+\beta)}{cos(\alpha)cos(\beta)}$

2) Use the identity sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

$\frac{sin\alpha.cos\beta +cos\alpha.sin\beta}{cos\alpha.cos \beta}$

3) Distributive property of division:

$\frac{sin\alpha.cos\beta}{cos\alpha.cos \beta}+\frac{cos\alpha.sin\beta}{cos\alpha.cos \beta}$

4) Simplify common factors in numerators and denominators

$\frac{sin\alpha}{cos\alpha }+\frac{sin\beta}{cos \beta}$

5) Definition of tangent ratio

tanα + tanβ

6) Conclusion

$\frac{sin\alpha.cos\beta +cos\alpha.sin\beta}{cos\alpha.cos \beta}=tan\alpha +tan\beta$