Certainly! Let's analyze and simplify the given trigonometric expression step by step.
Given expression:
[tex]\[ 2 \cos^2 \frac{x}{2} - 1 \][/tex]
### Step 1: Recall the Double Angle Identity for Cosine
To simplify this expression, we use one of the double angle identities for cosine. The double angle identity states:
[tex]\[ \cos(2\theta) = 2 \cos^2 \theta - 1 \][/tex]
### Step 2: Apply the Identity
In the given expression, our angle is [tex]\(\frac{x}{2}\)[/tex]. To use the double angle identity, let [tex]\(\theta = \frac{x}{2}\)[/tex].
### Step 3: Substitute [tex]\(\theta\)[/tex] in the Identity
Substituting [tex]\(\theta = \frac{x}{2}\)[/tex] into the double angle identity, we have:
[tex]\[ \cos(2 \cdot \frac{x}{2}) = 2 \cos^2 \frac{x}{2} - 1 \][/tex]
### Step 4: Simplify
Notice that [tex]\(2 \cdot \frac{x}{2} = x\)[/tex]. So, the expression simplifies to:
[tex]\[ \cos(x) = 2 \cos^2 \frac{x}{2} - 1 \][/tex]
### Step 5: Conclude the Simplified Expression
From our previous steps, we can directly conclude that:
[tex]\[ 2 \cos^2 \frac{x}{2} - 1 = \cos(x) \][/tex]
Therefore, the simplified form of the given expression [tex]\( 2 \cos^2 \frac{x}{2} - 1 \)[/tex] is:
[tex]\[ \cos(x) \][/tex]
This completes our step-by-step simplification of the given trigonometric expression.
