To determine which of the given expressions could not be used to find the exact value of [tex]\(\cos 90^\circ\)[/tex], we need to evaluate each expression separately. We know that [tex]\(\cos 90^\circ = 0\)[/tex]. Therefore, expressions that do not evaluate to 0 cannot be used to determine [tex]\(\cos 90^\circ\)[/tex].
First, recall some trigonometric identities and exact values:
- [tex]\(\sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)[/tex]
- Trigonometric identity: [tex]\(\cos^2 \theta + \sin^2 \theta = 1\)[/tex]
Now, we evaluate each given expression:
### Expression 1: [tex]\(\cos^2(45^\circ) - \sin^2(45^\circ)\)[/tex]
Given:
[tex]\[
\cos^2(45^\circ) - \sin^2(45^\circ)
\][/tex]
Substitute [tex]\(\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}\)[/tex]:
[tex]\[
\left(\frac{\sqrt{2}}{2}\right)^2 - \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} - \frac{2}{4} = 0
\][/tex]
Thus,
[tex]\[
\cos^2(45^\circ) - \sin^2(45^\circ) = 0
\][/tex]
### Expression 2: [tex]\(2 \sin(45^\circ) \cos(45^\circ)\)[/tex]
Given:
[tex]\[
2 \sin(45^\circ) \cos(45^\circ)
\][/tex]
Substitute [tex]\(\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}\)[/tex]:
[tex]\[
2 \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) = 2 \cdot \frac{2}{4} = 1
\][/tex]
Thus,
[tex]\[
2 \sin(45^\circ) \cos(45^\circ) = 1
\][/tex]
### Expression 3: [tex]\(2 \cos^2(45^\circ) - 1\)[/tex]
Given:
[tex]\[
2 \cos^2(45^\circ) - 1
\][/tex]
Substitute [tex]\(\cos 45^\circ = \frac{\sqrt{2}}{2}\)[/tex]:
[tex]\[
2 \left(\frac{\sqrt{2}}{2}\right)^2 - 1 = 2 \cdot \frac{2}{4} - 1 = 1 - 1 = 0
\][/tex]
Thus,
[tex]\[
2 \cos^2(45^\circ) - 1 = 0
\][/tex]
### Expression 4: [tex]\(1 - 2 \sin^2(45^\circ)\)[/tex]
Given:
[tex]\[
1 - 2 \sin^2(45^\circ)
\][/tex]
Substitute [tex]\(\sin 45^\circ = \frac{\sqrt{2}}{2}\)[/tex]:
[tex]\[
1 - 2 \left(\frac{\sqrt{2}}{2}\right)^2 = 1 - 2 \cdot \frac{2}{4} = 1 - 1 = 0
\][/tex]
Thus,
[tex]\[
1 - 2 \sin^2(45^\circ) = 0
\][/tex]
### Conclusion
From the evaluations above, we see which expressions match [tex]\(\cos 90^\circ = 0\)[/tex]:
- [tex]\(\cos^2(45^\circ) - \sin^2(45^\circ) = 0\)[/tex]
- [tex]\(2 \sin(45^\circ) \cos(45^\circ) = 1\)[/tex]
- [tex]\(2 \cos^2(45^\circ) - 1 = 0\)[/tex]
- [tex]\(1 - 2 \sin^2(45^\circ) = 0\)[/tex]
Therefore, the expression that does not match [tex]\(\cos 90^\circ = 0\)[/tex] is:
[tex]\[
2 \sin (45^\circ) \cos (45^\circ)
\][/tex]
