To simplify the expression [tex]\(2 \cos^2 5\theta - 1\)[/tex], we can use one of the trigonometric double-angle formulas. Specifically, we will use the double-angle formula for cosine:
[tex]\[ \cos(2x) = 2 \cos^2(x) - 1 \][/tex]
In this formula, [tex]\(x\)[/tex] represents the angle, and [tex]\(2x\)[/tex] is double that angle.
First, let's identify the angle in the given expression [tex]\(2 \cos^2 5\theta - 1\)[/tex]. Here, [tex]\(x = 5\theta\)[/tex].
Now substitute [tex]\(x\)[/tex] with [tex]\(5\theta\)[/tex] in the double-angle formula:
[tex]\[ \cos(2(5\theta)) = 2 \cos^2(5\theta) - 1 \][/tex]
This simplifies to:
[tex]\[ \cos(10\theta) = 2 \cos^2(5\theta) - 1 \][/tex]
Thus, [tex]\(2 \cos^2(5\theta) - 1\)[/tex] directly corresponds to [tex]\(\cos(10\theta)\)[/tex].
Therefore, the simplified form of the expression [tex]\(2 \cos^2 5\theta - 1\)[/tex] is:
[tex]\[ \cos(10\theta) \][/tex]
