If `4cos^(2)x=3 and x` is an acute angle, find the value of :
`cos3x`

Answer

To solve the problem, we will follow these steps:

### Step 1: Start with the given equation
We are given the equation:
\[ 4 \cos^2 x = 3 \]

### Step 2: Isolate \(\cos^2 x\)
To isolate \(\cos^2 x\), we divide both sides of the equation by 4:
\[ \cos^2 x = \frac{3}{4} \]

### Step 3: Take the square root
Next, we take the square root of both sides. Since \(x\) is an acute angle, we consider the positive root:
\[ \cos x = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Step 4: Identify the angle
We know that:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]
Thus, we can conclude:
\[ x = 30^\circ \]

### Step 5: Find \(3x\)
Now, we need to find \(3x\):
\[ 3x = 3 \times 30^\circ = 90^\circ \]

### Step 6: Calculate \(\cos 3x\)
Finally, we calculate \(\cos 3x\):
\[ \cos 3x = \cos 90^\circ = 0 \]

### Final Answer
Thus, the value of \(\cos 3x\) is:
\[ \cos 3x = 0 \]

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