Evaluate :
`tan^(-1) (2 cos ( 2 sin^(-1) (1/2)))`

Answer

To evaluate the expression \( \tan^{-1} \left( 2 \cos \left( 2 \sin^{-1} \left( \frac{1}{2} \right) \right) \right) \), we will follow these steps:

### Step 1: Evaluate \( \sin^{-1} \left( \frac{1}{2} \right) \)
The value of \( \sin^{-1} \left( \frac{1}{2} \right) \) is \( \frac{\pi}{6} \) because \( \sin \left( \frac{\pi}{6} \right) = \frac{1}{2} \).

### Step 2: Multiply by 2
Now we calculate \( 2 \sin^{-1} \left( \frac{1}{2} \right) \):
\[ 2 \sin^{-1} \left( \frac{1}{2} \right) = 2 \times \frac{\pi}{6} = \frac{\pi}{3} \]

### Step 3: Evaluate \( \cos \left( \frac{\pi}{3} \right) \)
Next, we find \( \cos \left( \frac{\pi}{3} \right) \):
\[ \cos \left( \frac{\pi}{3} \right) = \frac{1}{2} \]

### Step 4: Multiply by 2
Now we compute \( 2 \cos \left( \frac{\pi}{3} \right) \):
\[ 2 \cos \left( \frac{\pi}{3} \right) = 2 \times \frac{1}{2} = 1 \]

### Step 5: Evaluate \( \tan^{-1}(1) \)
Finally, we find \( \tan^{-1}(1) \):
\[ \tan^{-1}(1) = \frac{\pi}{4} \]

### Final Answer
Thus, the value of the expression \( \tan^{-1} \left( 2 \cos \left( 2 \sin^{-1} \left( \frac{1}{2} \right) \right) \right) \) is:
\[ \frac{\pi}{4} \]

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