If ` 4 cos ^(2) A -3 =0 and 0^(@) le A le 90 ^(@) ` find :
cos 3A

Answer

To solve the problem, we need to find the value of \( \cos 3A \) given the equation \( 4 \cos^2 A - 3 = 0 \) and the condition \( 0^\circ \leq A \leq 90^\circ \).

### Step-by-step Solution:

1. **Start with the given equation:**
\[ 4 \cos^2 A - 3 = 0 \]

2. **Rearrange the equation to isolate \( \cos^2 A \):**
\[ 4 \cos^2 A = 3 \]

3. **Divide both sides by 4:**
\[ \cos^2 A = \frac{3}{4} \]

4. **Take the square root of both sides to find \( \cos A \):**
\[ \cos A = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

5. **Determine the angle \( A \):**
Since \( \cos A = \frac{\sqrt{3}}{2} \), we know that:
\[ A = 30^\circ \]

6. **Now, we need to find \( \cos 3A \):**
\[ 3A = 3 \times 30^\circ = 90^\circ \]

7. **Calculate \( \cos 90^\circ \):**
\[ \cos 90^\circ = 0 \]

### Final Answer:
\[ \cos 3A = 0 \]