If `((1+cosA))/(2)=x` then the value of x is

Options

A. `sin^(2)((A)/(2))`
B. `sqrt(sin((A)/(2)))`
C. `sqrt(cos((A)/(2)))`
D. `cos^(2)((A)/(2))`

Answer

To solve the equation \(\frac{1 + \cos A}{2} = x\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the equation:
\[ \frac{1 + \cos A}{2} = x \]

### Step 2: Use the identity for \(\cos A\)
We know that:
\[ 1 + \cos A = 2 \cos^2\left(\frac{A}{2}\right) \]
This is derived from the double angle formula for cosine.

### Step 3: Substitute the identity into the equation
Substituting the identity into our equation gives:
\[ \frac{2 \cos^2\left(\frac{A}{2}\right)}{2} = x \]

### Step 4: Simplify the equation
The \(2\) in the numerator and denominator cancels out:
\[ \cos^2\left(\frac{A}{2}\right) = x \]

### Conclusion
Thus, the value of \(x\) is:
\[ x = \cos^2\left(\frac{A}{2}\right) \]

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