Let `cos(2 tan^(-1) x)=1/2` then the value of x is

Options

A. `sqrt(3)`
B. `(1)/sqrt(3)`
C. `1-sqrt(3)`
D. `1-(1)/sqrt(3)`

Answer

To solve the equation \( \cos(2 \tan^{-1} x) = \frac{1}{2} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \cos(2 \tan^{-1} x) = \frac{1}{2} \]

### Step 2: Use the inverse cosine function
We can express this in terms of the inverse cosine:
\[ 2 \tan^{-1} x = \cos^{-1} \left( \frac{1}{2} \right) \]

### Step 3: Find the angle corresponding to \( \cos^{-1} \left( \frac{1}{2} \right) \)
The value of \( \cos^{-1} \left( \frac{1}{2} \right) \) is known:
\[ \cos^{-1} \left( \frac{1}{2} \right) = \frac{\pi}{3} \]
So, we can substitute this back into our equation:
\[ 2 \tan^{-1} x = \frac{\pi}{3} \]

### Step 4: Divide both sides by 2
Now, we divide both sides by 2 to isolate \( \tan^{-1} x \):
\[ \tan^{-1} x = \frac{\pi}{6} \]

### Step 5: Convert back from inverse tangent
To find \( x \), we take the tangent of both sides:
\[ x = \tan \left( \frac{\pi}{6} \right) \]

### Step 6: Calculate the tangent value
The value of \( \tan \left( \frac{\pi}{6} \right) \) is:
\[ \tan \left( \frac{\pi}{6} \right) = \frac{1}{\sqrt{3}} \]

### Final Answer
Thus, the value of \( x \) is:
\[ x = \frac{1}{\sqrt{3}} \]

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