If `e^(x)=y+sqrt(1+y^(2)` then the value of y is

Options

A. `e^(x)-e^(-x)`
B. `1/2(e^(x)-e^(-x))`
C. `e^(x)+e^(-x)`
D. `1/2(e^(x)+e^(-x))`

Answer

To solve the equation \( e^x = y + \sqrt{1 + y^2} \) for \( y \), we can follow these steps:

### Step 1: Rearrange the equation
We start with the given equation:
\[ e^x = y + \sqrt{1 + y^2} \]
We can rearrange this to isolate the square root:
\[ \sqrt{1 + y^2} = e^x - y \]

### Step 2: Square both sides
Next, we square both sides to eliminate the square root:
\[ 1 + y^2 = (e^x - y)^2 \]
Expanding the right side, we get:
\[ 1 + y^2 = e^{2x} - 2e^x y + y^2 \]

### Step 3: Simplify the equation
Now, we can simplify the equation by subtracting \( y^2 \) from both sides:
\[ 1 = e^{2x} - 2e^x y \]

### Step 4: Rearrange to solve for \( y \)
Next, we rearrange the equation to isolate the term involving \( y \):
\[ 2e^x y = e^{2x} - 1 \]
Now, divide both sides by \( 2e^x \):
\[ y = \frac{e^{2x} - 1}{2e^x} \]

### Step 5: Further simplify
We can simplify this expression:
\[ y = \frac{1}{2} \left( \frac{e^{2x}}{e^x} - \frac{1}{e^x} \right) \]
This simplifies to:
\[ y = \frac{1}{2} \left( e^x - e^{-x} \right) \]

### Final Answer
Thus, the value of \( y \) is:
\[ y = \frac{1}{2} (e^x - e^{-x}) \]

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