Let's critically review the proof step by step:
1. Step 1:
[tex]\[
\cos (2x) = 2 \cos^2(x) - 1
\][/tex]
This is the double angle formula for cosine and is correct.
2. Step 2:
[tex]\[
\text{Let } 2x = \theta.
\][/tex]
This substitution is correct for simplifying the notation later on.
3. Step 3:
[tex]\[
\text{Then } x = \frac{\theta}{2}.
\][/tex]
This directly follows from the substitution in Step 2 and is correct.
4. Step 4:
[tex]\[
\cos(\theta) = 2 \cos^2\left(\frac{\theta}{2}\right) - 1
\][/tex]
We replace [tex]\( x \)[/tex] by [tex]\(\frac{\theta}{2}\)[/tex] in the double angle formula. This step is correct.
5. Step 5:
[tex]\[
-1 + \cos(\theta) = 2 \cos^2\left(\frac{\theta}{2}\right)
\][/tex]
By isolating [tex]\( \cos^2\left(\frac{\theta}{2}\right) \)[/tex] on one side of the equation, this step is correct.
6. Step 6:
[tex]\[
\frac{1 + \cos(\theta)}{2} = \cos^2\left(\frac{\theta}{2}\right)
\][/tex]
We solve for [tex]\( \cos^2\left(\frac{\theta}{2}\right) \)[/tex] by dividing both sides by 2. This is correct.
7. Step 7:
[tex]\[
\cos^2\left(\frac{\theta}{2}\right) = \frac{1 + \cos(\theta)}{2}
\][/tex]
This step rephrases the previous equation and is correct.
8. Step 8:
[tex]\[
\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}
\][/tex]
By taking the square root of both sides, we include both the positive and negative roots. This step is correct.
Therefore, after reviewing each step, it turns out that none of the steps contain a mathematical error. All steps are mathematically valid and correctly derived.
