Let’s determine which expression is equivalent to [tex]\(3 \cos^2\left(\frac{\theta}{2}\right) - 3\)[/tex].
First, let's simplify the given expression:
[tex]\[ 3 \cos^2\left(\frac{\theta}{2}\right) - 3 \][/tex]
We can use the double-angle identity for cosine, which states:
[tex]\[ \cos^2\left(\frac{\theta}{2}\right) = \frac{1 + \cos(\theta)}{2} \][/tex]
Substitute this identity into the given expression:
[tex]\[ 3 \cos^2\left(\frac{\theta}{2}\right) - 3 = 3 \left(\frac{1 + \cos(\theta)}{2}\right) - 3 \][/tex]
Simplify inside the parentheses:
[tex]\[ = 3 \cdot \frac{1 + \cos(\theta)}{2} - 3 \][/tex]
[tex]\[ = \frac{3 (1 + \cos(\theta))}{2} - 3 \][/tex]
Now, distribute the 3:
[tex]\[ = \frac{3 + 3 \cos(\theta)}{2} - 3 \][/tex]
Rewrite the fraction:
[tex]\[ = \frac{3 + 3 \cos(\theta)}{2} - \frac{6}{2} \][/tex]
Combine the terms into a single fraction:
[tex]\[ = \frac{3 + 3 \cos(\theta) - 6}{2} \][/tex]
[tex]\[ = \frac{3 \cos(\theta) - 3}{2} \][/tex]
Simplify the numerator:
[tex]\[ = \frac{3 (\cos(\theta) - 1)}{2} \][/tex]
Thus, the simplified expression for [tex]\(3 \cos^2\left(\frac{\theta}{2}\right) - 3\)[/tex] is:
[tex]\[ \frac{3}{2} (\cos(\theta) - 1) \][/tex]
Among the given options, this corresponds to:
[tex]\[ \frac{3}{2} (\cos(\theta) - 1) \][/tex]
Therefore, the equivalent expression is:
[tex]\[ \frac{3}{2} (\cos(\theta) - 1) \][/tex]
