Let's determine if [tex]\(\frac{\pi}{12}\)[/tex] is a solution to the given equation:
First, we start with the equation:
[tex]\[ 2 \cos^2(4x) - 1 = 0 \][/tex]
We need to check if [tex]\(\frac{\pi}{12}\)[/tex] satisfies this equation. Substitute [tex]\(x\)[/tex] with [tex]\(\frac{\pi}{12}\)[/tex]:
[tex]\[ x = \frac{\pi}{12} \][/tex]
Now, substitute [tex]\(x\)[/tex] in the equation:
[tex]\[ 2 \cos^2(4 \cdot \frac{\pi}{12}) - 1 = 0 \][/tex]
Simplify inside the cosine function:
[tex]\[ 4 \cdot \frac{\pi}{12} = \frac{4\pi}{12} = \frac{\pi}{3} \][/tex]
So the equation becomes:
[tex]\[ 2 \cos^2\left(\frac{\pi}{3}\right) - 1 = 0 \][/tex]
Next, evaluate the cosine term:
[tex]\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \][/tex]
Now, substitute [tex]\(\cos\left(\frac{\pi}{3}\right)\)[/tex] back into the equation:
[tex]\[ 2 \left(\frac{1}{2}\right)^2 - 1 = 0 \][/tex]
Simplify the expression:
[tex]\[ 2 \cdot \frac{1}{4} - 1 = 0 \][/tex]
[tex]\[ \frac{2}{4} - 1 = 0 \][/tex]
[tex]\[ \frac{1}{2} - 1 = 0 \][/tex]
[tex]\[ -\frac{1}{2} = 0 \][/tex]
This equation is false. Therefore, [tex]\(\frac{\pi}{12}\)[/tex] does not satisfy the given equation.
Hence, the correct answer is:
[tex]\[ \boxed{\text{B. False}} \][/tex]
