Mathematics: 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

4x - 2 = 3x + 1Step-by-step explanation: sh papi...

Verify the identity. StartFraction sine (alpha plus

Ответ:

SAMANTHA1982

24.01.2022

Mathematics

(A)Quotient Identity

Step-by-step explanation:

To Prove: $\dfrac{\sin \left(\alpha +\beta \right)}{\cos \left(\alpha +\beta \right)}=\tan \left(\alpha +\beta \right)$

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

$\dfrac{\sin \left(\alpha +\beta \right)}{\cos \left(\alpha +\beta \right)}=\dfrac{\sin \alpha \cos \beta +\cos \alpha \sin \beta }{\cos \alpha \cos \beta -\sin \alpha \sin \beta }$

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

$=\dfrac{\frac{\sin \alpha \cos \beta +\cos \alpha \sin \beta }{\cos \alpha \cos \beta }}{\frac{\cos \alpha \cos \beta -\sin \alpha \sin \beta }{\cos \alpha \cos \beta }}$

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

$=\dfrac{\frac{\sin \alpha{\cos \beta }}{\cos \alpha{\cos \beta }}+\frac{{\cos \alpha }\sin \beta }{{\cos \alpha }\cos \beta }}{\frac{{\cos \alpha }{\cos \beta }}{{\cos \alpha }{\cos \beta }}-\frac{\sin \alpha \sin \beta }{\cos \alpha \cos \beta }}$

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

$=\dfrac{\frac{\sin \alpha }{\cos \alpha }+\frac{\sin \beta }{\cos \beta }}{1-\frac{\sin \alpha \sin \beta }{\cos \alpha \cos \beta }}$

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

$=\dfrac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }=\tan \left(\alpha +\beta \right)$

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