Question

verify the identity.
\\( \frac { \sin ( \alpha + \beta ) } { \cos \alpha \cos \beta } = \tan \alpha + \tan \beta \\)
rewrite the numerator on the left side of the identity using one of the sum and difference formulas.
\\( \square \cos \alpha \cos \beta \\)
rewrite the fraction from the previous step such that it is a sum or difference of two expressions. do not simplify the
result.
\\( \square \\)
divide out any common factors in the expression from the previous step.
\\( \square \\)
the expression from the previous step then simplifies to \\( \tan \alpha + \tan \beta \\) using what?
\\( \bigcirc \\) a. reciprocal identity
\\( \bigcirc \\) b. quotient identity
\\( \bigcirc \\) c. pythagorean identity

Explanation:

Step1: Use the sum formula for sine

$$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$$
So, \(\frac{\sin(\alpha + \beta)}{\cos\alpha\cos\beta}=\frac{\sin\alpha\cos\beta+\cos\alpha\sin\beta}{\cos\alpha\cos\beta}\)

Step2: Divide each term in the numerator by the denominator

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

Step3: Simplify each fraction

For \(\frac{\sin\alpha\cos\beta}{\cos\alpha\cos\beta}\), divide out the common factor \(\cos\beta\) to get \(\frac{\sin\alpha}{\cos\alpha}\). For \(\frac{\cos\alpha\sin\beta}{\cos\alpha\cos\beta}\), divide out the common factor \(\cos\alpha\) to get \(\frac{\sin\beta}{\cos\beta}\)

Step4: Use the quotient identity

Since \(\tan x=\frac{\sin x}{\cos x}\), then \(\frac{\sin\alpha}{\cos\alpha}+\frac{\sin\beta}{\cos\beta}=\tan\alpha+\tan\beta\)

Answer:

The first blank is filled with the sum formula for sine. The second blank is filled with dividing out common factors \(\cos\beta\) and \(\cos\alpha\). The third blank is filled with the quotient identity (Option B).