Suppose cos A is given. If only one value of cos `(A/2)` is possible, then A must be:

Options

A. An odd multiple of `90^@`
B. A multiple of `90^@`
C. An odd multiple of `180^@`
D. A multiple of `180^@`

Answer

To solve the problem, we need to determine the conditions under which only one value of \( \cos\left(\frac{A}{2}\right) \) is possible given \( \cos A \).

### Step-by-Step Solution:

1. **Understanding the Range of Cosine Function**:
The cosine function \( \cos A \) can take values from -1 to 1. The half-angle formula states that:
\[ \cos\left(\frac{A}{2}\right) = \sqrt{\frac{1 + \cos A}{2}} \quad \text{or} \quad \cos\left(\frac{A}{2}\right) = -\sqrt{\frac{1 + \cos A}{2}} \]
This means that for each angle \( A \), there can be two possible values for \( \cos\left(\frac{A}{2}\right) \).

2. **Condition for a Unique Value**:
For \( \cos\left(\frac{A}{2}\right) \) to have only one value, the expression \( \frac{1 + \cos A}{2} \) must equal 0. This occurs when:
\[ 1 + \cos A = 0 \implies \cos A = -1 \]

3. **Finding the Corresponding Angle A**:
The cosine of an angle is -1 at specific angles. The general solution for \( \cos A = -1 \) is:
\[ A = (2n + 1) \cdot 180^\circ \quad \text{for any integer } n \]
This means that \( A \) must be an odd multiple of \( 180^\circ \).

4. **Conclusion**:
Therefore, if only one value of \( \cos\left(\frac{A}{2}\right) \) is possible, \( A \) must be an odd multiple of \( 180^\circ \).

### Final Answer:
The correct option is: **Option 3: An odd multiple of 180 degrees.**