Question

Verify the identity. StartFraction sine (alpha plus beta )Over cosine alpha cosine beta EndFraction equals tangent alpha plus tangent beta Rewrite the numerator on the left side of the identity using one of the sum and difference formulas. StartFraction nothing Over cosine alpha cosine beta EndFraction Rewrite the fraction from the previous step such that it is a sum or difference of two expressions. Do not simplify the result. nothing Divide out any common factors in the expression from the previous step. nothing The expression from the previous step then simplifies to tangent alpha plus tangent beta using​ what? A. Quotient Identity B. Reciprocal Identity C. ​Even-Odd Identity D. Pythagorean Identity

Answer 1

(A)Quotient Identity

Explanation:

To Prove:
`(\sin \left(\alpha +\beta \right))/(\cos \left(\alpha +\beta \right))=\tan \left(\alpha +\beta \right)`

Rewrite the numerator on the left side of the identity using the sum and difference formulas.

`(\sin \left(\alpha +\beta \right))/(\cos \left(\alpha +\beta \right))=(\sin \alpha \cos \beta +\cos \alpha \sin \beta )/(\cos \alpha \cos \beta -\sin \alpha \sin \beta )`

Next, Divide the numerator and denominator by
`cos\alpha \text{cos}\beta`

`=((\sin \alpha \cos \beta +\cos \alpha \sin \beta )/(\cos \alpha \cos \beta ))/((\cos \alpha \cos \beta -\sin \alpha \sin \beta )/(\cos \alpha \cos \beta ))`

Rewrite the fraction from the previous step such that it is a sum or difference of two expressions as shown below

`=\frac{\frac{\sin \alpha{\cos \beta }}{\cos \alpha{\cos \beta }}+\frac{{\cos \alpha }\sin \beta }{{\cos \alpha }\cos \beta }}{\frac{{\cos \alpha }{\cos \beta }}{{\cos \alpha }{\cos \beta }}-(\sin \alpha \sin \beta )/(\cos \alpha \cos \beta )}`

Divide out any common factors in the expression from the previous step.

`=((\sin \alpha )/(\cos \alpha )+(\sin \beta )/(\cos \beta ))/(1-(\sin \alpha \sin \beta )/(\cos \alpha \cos \beta ))`

The expression from the previous step then simplifies to
`\tan \left(\alpha +\beta \right)` using​ the Quotient Identity

`=(\tan \alpha +\tan \beta )/(1-\tan \alpha \tan \beta )=\tan \left(\alpha +\beta \right)`