If `sin A = (1)/(2)` then what is the value of (cot A -cos A)?

Answer

To solve the problem, we need to find the value of \( \cot A - \cos A \) given that \( \sin A = \frac{1}{2} \).

### Step-by-Step Solution:

1. **Identify the values of sine, cosine, and cotangent:**
Given \( \sin A = \frac{1}{2} \), we can represent this in a right triangle where:
- The opposite side (perpendicular) to angle A is \( 1 \) (since sine is opposite/hypotenuse).
- The hypotenuse is \( 2 \).

2. **Use the Pythagorean theorem to find the adjacent side:**
According to the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \]
Plugging in the known values:
\[ 2^2 = 1^2 + \text{adjacent}^2 \]
\[ 4 = 1 + \text{adjacent}^2 \]
\[ \text{adjacent}^2 = 4 - 1 = 3 \]
\[ \text{adjacent} = \sqrt{3} \]

3. **Calculate \( \cot A \) and \( \cos A \):**
- \( \cot A = \frac{\text{adjacent}}{\text{opposite}} = \frac{\sqrt{3}}{1} = \sqrt{3} \)
- \( \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2} \)

4. **Substitute values into the expression \( \cot A - \cos A \):**
\[ \cot A - \cos A = \sqrt{3} - \frac{\sqrt{3}}{2} \]

5. **Simplify the expression:**
To subtract these, we need a common denominator:
\[ \sqrt{3} = \frac{2\sqrt{3}}{2} \]
Thus,
\[ \cot A - \cos A = \frac{2\sqrt{3}}{2} - \frac{\sqrt{3}}{2} = \frac{2\sqrt{3} - \sqrt{3}}{2} = \frac{\sqrt{3}}{2} \]

### Final Answer:
\[ \cot A - \cos A = \frac{\sqrt{3}}{2} \]