Let's carefully review the steps provided in the proof of the identity [tex]\(\sin \left(A - \frac{3\pi}{2}\right) = \cos A\)[/tex]:
Identity to prove:
[tex]\[
\sin \left(A - \frac{3\pi}{2}\right) = \cos A
\][/tex]
Step-by-step review:
1. Step 1:
[tex]\[
\sin \left(A - \frac{3\pi}{2}\right) = \sin A \cos \left(\frac{3\pi}{2}\right) - \cos A \sin \left(\frac{3\pi}{2}\right)
\][/tex]
This step applies the angle subtraction formula for sine correctly:
[tex]\[
\sin(A - B) = \sin A \cos B - \cos A \sin B
\][/tex]
2. Step 2:
[tex]\[
= (\sin A)(0) + (\cos A)(-1)
\][/tex]
Because:
[tex]\[
\cos \left(\frac{3\pi}{2}\right) = 0 \quad \text{and} \quad \sin \left(\frac{3\pi}{2}\right) = -1
\][/tex]
This step correctly substitutes the values of [tex]\(\cos \left(\frac{3\pi}{2}\right) = 0\)[/tex] and [tex]\(\sin \left(\frac{3\pi}{2}\right) = -1\)[/tex].
3. Step 3:
[tex]\[
= (\sin A)(0) + (1)(\cos A)
\][/tex]
Here's the error. The correct term should be:
[tex]\[
= (\sin A)(0) + (-1)(\cos A)
\][/tex]
Equivalently:
[tex]\[
= 0 - \cos A
\][/tex]
4. Step 4:
[tex]\[
0 + \cos A
\][/tex]
This step carries forward the error from Step 3. The correct simplification from Step 2 should have been:
[tex]\[
0 - \cos A
\][/tex]
5. Step 5:
[tex]\[
\cos A
\][/tex]
Again, this carries forward the mistake.
The error is first made in Step 3, where the negative sign in front of [tex]\(\cos A\)[/tex] is incorrectly considered as positive. Therefore, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
