Rewrite the trigonometric identity involving cosine.

Given:
[tex]\cos b=\frac{1}{2}[\sin (a+b)+\sin (a-b)][/tex]

Choose the correct option:

A. [tex]\cos b[/tex]
B. [tex]\cos a[/tex]
C. [tex]\sin a[/tex]
D. [tex]\sin b[/tex]



Answer :

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]