To prove that if [tex]\(\operatorname{Sin} \alpha + \operatorname{Sin} \beta = \frac{1}{4}\)[/tex] and [tex]\(\operatorname{Cos} \alpha + \operatorname{Cos} \beta = \frac{1}{2}\)[/tex], then [tex]\(\tan \left( \frac{\alpha + \beta}{2} \right) = \frac{1}{2}\)[/tex], follow these steps:
1. Set up expressions using sum-to-product identities:
Let [tex]\(A = \frac{\alpha + \beta}{2}\)[/tex] and [tex]\(B = \frac{\alpha - \beta}{2}\)[/tex]. Then we can use the sum-to-product identities:
[tex]\[
\operatorname{Sin} \alpha + \operatorname{Sin} \beta = 2 \operatorname{Sin}\left( \frac{\alpha + \beta}{2} \right) \operatorname{Cos}\left( \frac{\alpha - \beta}{2} \right) = 2 \operatorname{Sin}(A) \operatorname{Cos}(B)
\][/tex]
[tex]\[
\operatorname{Cos} \alpha + \operatorname{Cos} \beta = 2 \operatorname{Cos}\left( \frac{\alpha + \beta}{2} \right) \operatorname{Cos}\left( \frac{\alpha - \beta}{2} \right) = 2 \operatorname{Cos}(A) \operatorname{Cos}(B)
\][/tex]
2. Substitute the given values:
Given [tex]\(\operatorname{Sin} \alpha + \operatorname{Sin} \beta = \frac{1}{4}\)[/tex]:
[tex]\[
2 \operatorname{Sin}(A) \operatorname{Cos}(B) = \frac{1}{4}
\][/tex]
[tex]\[
\operatorname{Sin}(A) \operatorname{Cos}(B) = \frac{1}{8} \quad \text{(1)}
\][/tex]
Given [tex]\(\operatorname{Cos} \alpha + \operatorname{Cos} \beta = \frac{1}{2}\)[/tex]:
[tex]\[
2 \operatorname{Cos}(A) \operatorname{Cos}(B) = \frac{1}{2}
\][/tex]
[tex]\[
\operatorname{Cos}(A) \operatorname{Cos}(B) = \frac{1}{4} \quad \text{(2)}
\][/tex]
3. Divide equation (1) by equation (2):
[tex]\[
\frac{\operatorname{Sin}(A) \operatorname{Cos}(B)}{\operatorname{Cos}(A) \operatorname{Cos}(B)} = \frac{\frac{1}{8}}{\frac{1}{4}}
\][/tex]
[tex]\[
\frac{\operatorname{Sin}(A)}{\operatorname{Cos}(A)} = \frac{\frac{1}{8}}{\frac{1}{4}}
\][/tex]
[tex]\[
\frac{\operatorname{Sin}(A)}{\operatorname{Cos}(A)} = \frac{1}{2}
\][/tex]
4. Simplify to find [tex]\(\tan(A)\)[/tex]:
[tex]\[
\operatorname{Tan}(A) = \frac{1}{2}
\][/tex]
Thus, we have shown that:
[tex]\[
\tan \left( \frac{\alpha + \beta}{2} \right) = \frac{1}{2}
\][/tex]
