If `xepsilon(-1,1)` and `2tan^(-1)x=tan^(-1)y` then find `y` in term of `x`.

Explanation

To solve the equation \(2\tan^{-1}(x) = \tan^{-1}(y)\) and find \(y\) in terms of \(x\), we can follow these steps:

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

### Step 2: Use the double angle formula for tangent
Using the formula for the tangent of a double angle, we can express \(2\tan^{-1}(x)\) as:
\[ \tan(2\tan^{-1}(x)) = \frac{2\tan(\tan^{-1}(x))}{1 - \tan^2(\tan^{-1}(x))} \]
Since \(\tan(\tan^{-1}(x)) = x\), we have:
\[ \tan(2\tan^{-1}(x)) = \frac{2x}{1 - x^2} \]

### Step 3: Set the equation for \(y\)
Since we have \(2\tan^{-1}(x) = \tan^{-1}(y)\), we can equate the tangents:
\[ y = \tan(2\tan^{-1}(x)) = \frac{2x}{1 - x^2} \]

### Step 4: Final expression for \(y\)
Thus, we have found \(y\) in terms of \(x\):
\[ y = \frac{2x}{1 - x^2} \]

### Summary
The final result is:
\[ y = \frac{2x}{1 - x^2} \]

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