To determine the [tex]\(x\)[/tex]-values at which the graphs of the functions [tex]\(y = \cos(2x)\)[/tex] and [tex]\(y = \cos^2(x) - 1\)[/tex] intersect over the interval [tex]\(0 \leq x < 2\pi\)[/tex], we need to set the equations equal to each other and solve for [tex]\(x\)[/tex].
First, let's set the two functions equal to each other:
[tex]\[
\cos(2x) = \cos^2(x) - 1
\][/tex]
Next, let's use a trigonometric identity for [tex]\(\cos(2x)\)[/tex]:
[tex]\[
\cos(2x) = 2\cos^2(x) - 1
\][/tex]
Now, we can rewrite the equation substituting [tex]\(\cos(2x)\)[/tex]:
[tex]\[
2\cos^2(x) - 1 = \cos^2(x) - 1
\][/tex]
To simplify this, we subtract [tex]\((\cos^2(x) - 1)\)[/tex] from both sides:
[tex]\[
2\cos^2(x) - 1 - \cos^2(x) + 1 = 0
\][/tex]
This simplifies to:
[tex]\[
\cos^2(x) = 0
\][/tex]
Taking the square root of both sides, we get:
[tex]\[
\cos(x) = 0
\][/tex]
Now, we need to find the values of [tex]\(x\)[/tex] within the given interval [tex]\(0 \leq x < 2\pi\)[/tex] where [tex]\(\cos(x) = 0\)[/tex]. These values occur at:
[tex]\[
x = \frac{\pi}{2} \quad \text{and} \quad x = \frac{3\pi}{2}
\][/tex]
Both of these [tex]\(x\)[/tex]-values are within the interval [tex]\(0 \leq x < 2\pi\)[/tex], and thus, they are the points at which the graphs of the functions intersect.
Hence, the correct [tex]\(x\)[/tex]-values where the graphs of [tex]\(y = \cos(2x)\)[/tex] and [tex]\(y = \cos^2(x) - 1\)[/tex] intersect are:
[tex]\[
\boxed{\frac{\pi}{2}, \frac{3\pi}{2}}
\][/tex]
The options you should select are:
[tex]\(\frac{\pi}{2}\)[/tex]
[tex]\(\frac{3\pi}{2}\)[/tex]
