The simplest form of `(cos^(-1)x +cos^(-1){x/2+1/2sqrt(3-3x^2)}),AA x in [1/2,1]` is A. Then the value of tanA is

Options

A. `1sqrt3`
B. `sqrt3`
C. 1
D. -1

Explanation

To solve the problem step by step, we need to simplify the expression \( a = \cos^{-1} x + \cos^{-1} \left( \frac{x}{2} + \frac{1}{2} \sqrt{3 - 3x^2} \right) \) for \( x \in \left[ \frac{1}{2}, 1 \right] \) and then find \( \tan A \).

### Step 1: Substitute \( x = \cos \theta \)
Let \( x = \cos \theta \). Then, we can rewrite the expression for \( a \):
\[ a = \cos^{-1}(\cos \theta) + \cos^{-1} \left( \frac{\cos \theta}{2} + \frac{1}{2} \sqrt{3 - 3 \cos^2 \theta} \right) \]
Since \( \cos^{-1}(\cos \theta) = \theta \) for \( \theta \in [0, \pi] \), we have:
\[ a = \theta + \cos^{-1} \left( \frac{\cos \theta}{2} + \frac{1}{2} \sqrt{3 - 3 \cos^2 \theta} \right) \]

### Step 2: Simplify the second term
Now, we simplify the term \( \frac{1}{2} \sqrt{3 - 3 \cos^2 \theta} \):
\[ \sqrt{3 - 3 \cos^2 \theta} = \sqrt{3(1 - \cos^2 \theta)} = \sqrt{3 \sin^2 \theta} = \sqrt{3} \sin \theta \]
Thus, we can rewrite the expression as:
\[ a = \theta + \cos^{-1} \left( \frac{\cos \theta}{2} + \frac{\sqrt{3}}{2} \sin \theta \right) \]

### Step 3: Recognize the cosine of a sum
We recognize that:
\[ \frac{1}{2} = \cos \frac{\pi}{3} \quad \text{and} \quad \frac{\sqrt{3}}{2} = \sin \frac{\pi}{3} \]
This allows us to use the cosine addition formula:
\[ \cos \left( \frac{\pi}{3} - \theta \right) = \cos \frac{\pi}{3} \cos \theta + \sin \frac{\pi}{3} \sin \theta \]
Thus, we can write:
\[ \frac{\cos \theta}{2} + \frac{\sqrt{3}}{2} \sin \theta = \cos \left( \frac{\pi}{3} - \theta \right) \]

### Step 4: Substitute back into the expression for \( a \)
Now substituting back, we have:
\[ a = \theta + \cos^{-1} \left( \cos \left( \frac{\pi}{3} - \theta \right) \right) \]
Since \( \cos^{-1}(\cos x) = x \) for \( x \in [0, \pi] \), we get:
\[ a = \theta + \left( \frac{\pi}{3} - \theta \right) = \frac{\pi}{3} \]

### Step 5: Find \( \tan A \)
Now that we have \( a = \frac{\pi}{3} \), we can find \( \tan A \):
\[ \tan A = \tan \left( \frac{\pi}{3} \right) = \sqrt{3} \]

### Final Answer
Thus, the value of \( \tan A \) is:
\[ \boxed{\sqrt{3}} \]