If `0 lt x lt (pi)/(2)` and `tan x = (a)/(2)`, then cos x =

Options

A. `(2)/(sqrt(a^(2)-4))`
B. `(a)/(sqrt(a^(2)-4))`
C. `(2)/(a+2)`
D. `(2)/(sqrt(a^(2)+4))`

Explanation

To solve the problem, we need to find the value of \( \cos x \) given that \( \tan x = \frac{a}{2} \) and \( 0 < x < \frac{\pi}{2} \).

### Step-by-Step Solution:

1. **Understanding the Tangent Function**:
- We know that \( \tan x = \frac{\text{opposite}}{\text{adjacent}} \).
- Given \( \tan x = \frac{a}{2} \), we can denote the opposite side of the triangle as \( a \) and the adjacent side as \( 2 \).

2. **Drawing the Right Triangle**:
- Construct a right triangle where:
- The angle \( x \) is at one vertex.
- The side opposite to angle \( x \) is \( a \).
- The side adjacent to angle \( x \) is \( 2 \).

3. **Finding the Hypotenuse**:
- By the Pythagorean theorem, the hypotenuse \( h \) can be calculated as follows:
\[ h = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} = \sqrt{a^2 + 2^2} = \sqrt{a^2 + 4} \]

4. **Calculating \( \cos x \)**:
- The cosine of angle \( x \) is given by the ratio of the adjacent side to the hypotenuse:
\[ \cos x = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{2}{\sqrt{a^2 + 4}} \]

5. **Final Result**:
- Therefore, the value of \( \cos x \) is:
\[ \cos x = \frac{2}{\sqrt{a^2 + 4}} \]