Find the set of value of x for which the equation `cos^(-1) x + cos^(-1) ((x)/(2) + (1)/(2) sqrt(3 -3x^(2))) = (pi)/(3)` holds goods

Explanation

To solve the equation \( \cos^{-1} x + \cos^{-1} \left( \frac{x}{2} + \frac{1}{2} \sqrt{3 - 3x^2} \right) = \frac{\pi}{3} \), we will follow these steps:

### Step 1: Understand the equation
We have two inverse cosine terms that sum up to \( \frac{\pi}{3} \). We can use the property of inverse trigonometric functions to simplify our calculations.

### Step 2: Use the property of cosine
Recall that if \( \cos^{-1} a + \cos^{-1} b = \frac{\pi}{3} \), then:
\[ a = \cos\left(\frac{\pi}{3} - \cos^{-1} b\right) \]
This implies:
\[ x = \cos\left(\frac{\pi}{3} - \cos^{-1} \left( \frac{x}{2} + \frac{1}{2} \sqrt{3 - 3x^2} \right)\right) \]

### Step 3: Set up the equation
Using the cosine addition formula, we can express this as:
\[ x = \frac{1}{2} \left( \frac{x}{2} + \frac{1}{2} \sqrt{3 - 3x^2} \right) + \frac{\sqrt{3}}{2} \sin\left(\cos^{-1} \left( \frac{x}{2} + \frac{1}{2} \sqrt{3 - 3x^2} \right)\right) \]

### Step 4: Simplify the equation
To simplify, we can express \( \sin\left(\cos^{-1} y\right) \) as \( \sqrt{1 - y^2} \):
\[ y = \frac{x}{2} + \frac{1}{2} \sqrt{3 - 3x^2} \]
Thus, we need to calculate \( 1 - y^2 \).

### Step 5: Solve for \( x \)
We can rearrange and solve the resulting equation for \( x \). This may involve squaring both sides and simplifying further.

### Step 6: Find the range of \( x \)
Since \( \cos^{-1} \) is defined for \( x \in [-1, 1] \), we will check the values of \( x \) that satisfy this condition.

### Step 7: Verify solutions
Finally, we will substitute back the values of \( x \) into the original equation to verify if they satisfy it.

### Final Answer
The set of values of \( x \) that satisfy the equation can be determined through the above steps.

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