If `4cos^(2)x=3 and x` is an acute angle, find the value of :
`cos^(2)x+cot^(2)x`

Answer

To solve the problem, we need to find the value of \( \cos^2 x + \cot^2 x \) given that \( 4 \cos^2 x = 3 \) and \( x \) is an acute angle.

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ 4 \cos^2 x = 3 \]

2. **Isolate \( \cos^2 x \):**
\[ \cos^2 x = \frac{3}{4} \]

3. **Find \( \cos x \):**
\[ \cos x = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]
Since \( x \) is an acute angle, we take the positive root.

4. **Determine \( \cot x \):**
We know that:
\[ \cot x = \frac{\cos x}{\sin x} \]
To find \( \sin^2 x \), we use the identity \( \sin^2 x + \cos^2 x = 1 \):
\[ \sin^2 x = 1 - \cos^2 x = 1 - \frac{3}{4} = \frac{1}{4} \]
Thus,
\[ \sin x = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

5. **Calculate \( \cot^2 x \):**
\[ \cot x = \frac{\cos x}{\sin x} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \]
Therefore,
\[ \cot^2 x = (\sqrt{3})^2 = 3 \]

6. **Combine \( \cos^2 x \) and \( \cot^2 x \):**
\[ \cos^2 x + \cot^2 x = \frac{3}{4} + 3 \]
To add these, convert \( 3 \) into a fraction with a denominator of \( 4 \):
\[ 3 = \frac{12}{4} \]
Thus,
\[ \cos^2 x + \cot^2 x = \frac{3}{4} + \frac{12}{4} = \frac{15}{4} \]

### Final Answer:
\[ \cos^2 x + \cot^2 x = \frac{15}{4} \]