If `cos((A)/(2))=x` then the value of x is

Options

A. `sqrt([((1-cosA))/(2)])`
B. `sqrt([((1+sinA))/(2)])`
C. `sqrt([((1-sinA))/(2)])`
D. `sqrt([((1+cosA))/(2)])`

Answer

To solve the problem where \( \cos\left(\frac{A}{2}\right) = x \) and we need to find the value of \( x \), we can use the half-angle identity for cosine.

### Step-by-step Solution:

1. **Recall the Half-Angle Identity**:
The half-angle identity for cosine states that:
\[ \cos\left(\frac{A}{2}\right) = \sqrt{\frac{1 + \cos A}{2}} \]
This means that to find \( \cos\left(\frac{A}{2}\right) \), we need to know the value of \( \cos A \).

2. **Substituting the Identity**:
Given \( \cos\left(\frac{A}{2}\right) = x \), we can substitute the half-angle identity into this equation:
\[ x = \sqrt{\frac{1 + \cos A}{2}} \]

3. **Squaring Both Sides**:
To eliminate the square root, we square both sides of the equation:
\[ x^2 = \frac{1 + \cos A}{2} \]

4. **Rearranging the Equation**:
Multiply both sides by 2 to isolate \( 1 + \cos A \):
\[ 2x^2 = 1 + \cos A \]
Now, we can rearrange this to find \( \cos A \):
\[ \cos A = 2x^2 - 1 \]

5. **Conclusion**:
The value of \( x \) is determined by the cosine of half the angle \( A \). Thus, we have:
\[ x = \cos\left(\frac{A}{2}\right) = \sqrt{\frac{1 + \cos A}{2}} \]

### Final Answer:
The value of \( x \) is \( \sqrt{\frac{1 + \cos A}{2}} \).

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