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Which sum or difference identity would you use to verify that [tex]\sin \left(90^{\circ}+\theta\right)=\cos \theta[/tex]?

A. [tex]\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta[/tex]
B. [tex]\sin (\alpha+\beta)=\sin \alpha \cos \beta+\cos \alpha \sin \beta[/tex]
C. [tex]\sin (\alpha-\beta)=\sin \alpha \cos \beta-\cos \alpha \sin \beta[/tex]
D. [tex]\cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta[/tex]

Please select the best answer from the choices provided:
A
B
C
D

Sagot :

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