To verify the trigonometric identity [tex]\(\sin(90^\circ + \theta) = \cos \theta\)[/tex], we should use the sum identity for sine. Here is the step-by-step solution:
1. Identify the appropriate sum identity for sine:
We need the sum identity for sine because we are dealing with an expression of the form [tex]\(\sin(\alpha + \beta)\)[/tex].
2. Recall the sum identity for sine:
[tex]\(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\)[/tex].
3. Apply this identity to our specific case:
In our problem, [tex]\(\alpha = 90^\circ\)[/tex] and [tex]\(\beta = \theta\)[/tex]. So, we rewrite the identity as:
[tex]\[
\sin(90^\circ + \theta) = \sin 90^\circ \cos \theta + \cos 90^\circ \sin \theta
\][/tex]
4. Evaluate the trigonometric values at [tex]\(90^\circ\)[/tex]:
[tex]\[
\sin 90^\circ = 1 \quad \text{and} \quad \cos 90^\circ = 0
\][/tex]
5. Substitute these values into the identity:
[tex]\[
\sin(90^\circ + \theta) = 1 \cdot \cos \theta + 0 \cdot \sin \theta
\][/tex]
6. Simplify the expression:
[tex]\[
\sin(90^\circ + \theta) = \cos \theta
\][/tex]
Thus, we verified that [tex]\(\sin(90^\circ + \theta) = \cos \theta\)[/tex] using the sum identity for sine, which is:
[tex]\[
\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
\][/tex]
Therefore, the correct answer is:
b. [tex]\(\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\)[/tex]
So, the best answer from the choices provided is:
B
