To determine the correct match for the given trigonometric identity:
[tex]\[ \cos(b) = \frac{1}{2} [\sin(a + b) + \sin(a - b)] \][/tex]
we need to understand each option in the context of trigonometric identities and functions.
1. Trigonometric identity analysis:
Recall the sum and difference identities for sine:
[tex]\[
\sin(x + y) = \sin(x)\cos(y) + \cos(x)\sin(y)
\][/tex]
[tex]\[
\sin(x - y) = \sin(x)\cos(y) - \cos(x)\sin(y)
\][/tex]
Also, for cosine, an identity that might be handy is:
[tex]\[
\cos(x \pm y) = \cos(x)\cos(y) \mp \sin(x)\sin(y)
\][/tex]
2. Comparing sides:
Given trigonometric identity:
[tex]\[
\cos(b) = \frac{1}{2}[\sin(a + b) + \sin(a - b)]
\][/tex]
Let's check whether matching options apply:
A. Option A: [tex]\(\cos b\)[/tex]
The left-hand side of the given identity is [tex]\(\cos b\)[/tex].
B. Option B: [tex]\(\cos a\)[/tex]
There is no [tex]\(\cos a\)[/tex] on either side of the identity, so this cannot be correct.
C. Option C: [tex]\(\sin 8\)[/tex]
[tex]\(\sin 8\)[/tex] (where 8 presumably is an angle in degrees or radians) would not logically correspond within the context of the given identity.
D. Option D: [tex]\(\sin b\)[/tex]
[tex]\(\sin b\)[/tex] does not match with either the left-hand side ([tex]\(\cos b\)[/tex]) or the right-hand side formulations which involve sine sums.
3. Conclusion
After verifying each option:
- Only Option (A) corresponds directly to the left-hand side and follows the form and trigonometric identities given.
Therefore, the correct match for the given trigonometric identity:
[tex]\[ \cos(b) = \frac{1}{2}[\sin(a + b) + \sin(a - b)] \][/tex]
is:
A. [tex]\(\cos b\)[/tex]
