To determine the exact value of [tex]\(\cos 300^\circ\)[/tex], let us proceed by identifying the correct expression from the given options.
### Step-by-Step Solution:
1. Understanding [tex]\(\cos 300^\circ\)[/tex]:
[tex]\(\cos 300^\circ\)[/tex] is the cosine of an angle that is located in the fourth quadrant of the unit circle. The equivalent reference angle is [tex]\(360^\circ - 300^\circ = 60^\circ\)[/tex].
2. Using Properties of Cosine in the Fourth Quadrant:
In the fourth quadrant, cosine values are positive. Therefore,
[tex]\[
\cos 300^\circ = \cos(360^\circ - 60^\circ) = \cos 60^\circ.
\][/tex]
We know that [tex]\(\cos 60^\circ = \frac{1}{2}\)[/tex].
3. Examining the Given Options:
Let's analyze each expression:
- Option 1: [tex]\(1 - 2 \sin^2(150^\circ)\)[/tex]:
[tex]\(150^\circ\)[/tex] is in the second quadrant. The sine function is positive in the second quadrant, and [tex]\(\sin 150^\circ = \sin 30^\circ = \frac{1}{2}\)[/tex].
Plugging in the value, we get:
[tex]\[
1 - 2 \sin^2(150^\circ) = 1 - 2 \left(\frac{1}{2}\right)^2 = 1 - 2 \cdot \frac{1}{4} = 1 - \frac{1}{2} = \frac{1}{2}.
\][/tex]
This evaluates to [tex]\(\frac{1}{2}\)[/tex], which is equal to [tex]\(\cos 300^\circ\)[/tex].
- Option 2: [tex]\(2 \sin(150^\circ) \cos(150^\circ)\)[/tex]:
Plugging in the values, [tex]\(\sin 150^\circ = \frac{1}{2}\)[/tex] and [tex]\(\cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}\)[/tex]. Then,
[tex]\[
2 \sin(150^\circ) \cos(150^\circ) = 2 \cdot \frac{1}{2} \cdot \left(-\frac{\sqrt{3}}{2}\right) = -\frac{\sqrt{3}}{2}.
\][/tex]
This does not correspond to [tex]\(\cos 300^\circ\)[/tex], which should be [tex]\(\frac{1}{2}\)[/tex].
- Option 3: [tex]\(2 \cos^2(300^\circ) - 1\)[/tex]:
We use the cosine value of [tex]\(300^\circ\)[/tex], which is [tex]\(\cos 300^\circ = \frac{1}{2}\)[/tex]. Plugging in the value, we get:
[tex]\[
2 \cos^2(300^\circ) - 1 = 2 \left(\frac{1}{2}\right)^2 - 1 = 2 \cdot \frac{1}{4} - 1 = \frac{1}{2} - 1 = -\frac{1}{2}.
\][/tex]
This evaluates to [tex]\(\cos 300^\circ\)[/tex].
- Option 4: [tex]\(\cos^2(300^\circ) - 2 \sin(300^\circ)\)[/tex]:
We use [tex]\(\cos 300^\circ = \frac{1}{2}\)[/tex] and [tex]\(\sin 300^\circ = -\sin 60^\circ = -\frac{\sqrt{3}}{2}\)[/tex]. Plugging in the values, we get:
[tex]\[
\cos^2(300^\circ) - 2 \sin(300^\circ) = \left(\frac{1}{2}\right)^2 - 2 \left(-\frac{\sqrt{3}}{2}\right) = \frac{1}{4} + \sqrt{3}.
\][/tex]
This does not correspond to [tex]\(\cos 300^\circ\)[/tex].
After evaluating all the options, the correct expression that gives the exact value of [tex]\(\cos 300^\circ\)[/tex] is:
[tex]\[
2 \cos^2(300^\circ) - 1
\][/tex]
Thus, the correct option is:
[tex]\(\boxed{3}\)[/tex]
