To find the value of [tex]\(\cos \left(\frac{\alpha+\beta}{2}\right)\)[/tex], given that [tex]\(\cos (\alpha+\beta) = 0\)[/tex], we can follow these steps:
1. Recognize the Relationship: First, we need to understand the relationship given by [tex]\(\cos (\alpha+\beta) = 0\)[/tex]. The cosine function is zero when its argument is an odd multiple of [tex]\(\frac{\pi}{2}\)[/tex]:
[tex]\[
\alpha + \beta = \frac{\pi}{2} + k\pi
\][/tex]
where [tex]\(k\)[/tex] is any integer.
2. Simplification for Simplicity: Since [tex]\(\cos (\alpha+\beta)\)[/tex] is periodic with period [tex]\(2\pi\)[/tex], we can simplify our calculations by choosing the smallest positive multiple. This means:
[tex]\[
\alpha + \beta = \frac{\pi}{2}
\][/tex]
3. Halving the Angle: Now, we need to calculate [tex]\(\cos \left(\frac{\alpha+\beta}{2}\right)\)[/tex]. Substitute the value we have from step 2:
[tex]\[
\frac{\alpha + \beta}{2} = \frac{\frac{\pi}{2}}{2} = \frac{\pi}{4}
\][/tex]
4. Calculate the Cosine Value: Finally, compute the cosine of [tex]\(\frac{\pi}{4}\)[/tex]:
[tex]\[
\cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\][/tex]
In numerical form, the value of [tex]\(\frac{\sqrt{2}}{2}\)[/tex], which is a commonly known trigonometric value, is approximately:
[tex]\[
0.7071067811865476
\][/tex]
Therefore, the value of [tex]\(\cos \left(\frac{\alpha + \beta}{2}\right)\)[/tex] is:
[tex]\[
0.7071067811865476
\][/tex]
