If 2 sin `x = sqrt3` , evaluate .
`3 cos x - 4 cos^3x`.

Answer

To solve the problem, we start with the equation given:

**Step 1: Start with the equation**
Given:
\[ 2 \sin x = \sqrt{3} \]

**Step 2: Solve for sin x**
To isolate \(\sin x\), divide both sides by 2:
\[ \sin x = \frac{\sqrt{3}}{2} \]

**Step 3: Identify the angle x**
We know that \(\sin 60^\circ = \frac{\sqrt{3}}{2}\). Therefore, we can conclude:
\[ x = 60^\circ \]

**Step 4: Find cos x**
Next, we need to find \(\cos x\):
\[ \cos 60^\circ = \frac{1}{2} \]

**Step 5: Substitute cos x into the expression**
Now we substitute \(\cos x\) into the expression \(3 \cos x - 4 \cos^3 x\):
\[ 3 \cos x - 4 \cos^3 x = 3 \left(\frac{1}{2}\right) - 4 \left(\frac{1}{2}\right)^3 \]

**Step 6: Calculate each term**
Calculate \(3 \left(\frac{1}{2}\right)\):
\[ 3 \cdot \frac{1}{2} = \frac{3}{2} \]

Calculate \(4 \left(\frac{1}{2}\right)^3\):
\[ 4 \cdot \left(\frac{1}{2}\right)^3 = 4 \cdot \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \]

**Step 7: Combine the results**
Now, substitute these values back into the expression:
\[ \frac{3}{2} - \frac{1}{2} = \frac{3 - 1}{2} = \frac{2}{2} = 1 \]

**Final Answer:**
Thus, the value of \(3 \cos x - 4 \cos^3 x\) is:
\[ \boxed{1} \]

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