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

Options

A. `(1)/(2)(e^(x) + e^(-x))`
B. `(1)/(2)(e^(x)-e^(-x))`
C. `(e^(x)+e^(-x))`
D. none

Answer

To find the value of \( t \) given the equation \( e^x = t + \sqrt{1 + t^2} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ e^x = t + \sqrt{1 + t^2} \]

### Step 2: Isolate the square root
We can isolate the square root by rearranging the equation:
\[ \sqrt{1 + t^2} = e^x - t \]

### Step 3: Square both sides
Next, we square both sides to eliminate the square root:
\[ 1 + t^2 = (e^x - t)^2 \]

### Step 4: Expand the right-hand side
Expanding the right-hand side gives:
\[ 1 + t^2 = e^{2x} - 2te^x + t^2 \]

### Step 5: Simplify the equation
Now, we can simplify the equation by subtracting \( t^2 \) from both sides:
\[ 1 = e^{2x} - 2te^x \]

### Step 6: Rearrange to solve for \( t \)
Rearranging gives us:
\[ 2te^x = e^{2x} - 1 \]
\[ t = \frac{e^{2x} - 1}{2e^x} \]

### Step 7: Simplify the expression for \( t \)
We can simplify this expression:
\[ t = \frac{(e^x - 1)(e^x + 1)}{2e^x} = \frac{e^x - 1}{2} + \frac{e^x + 1}{2} \]

### Step 8: Recognize the hyperbolic sine function
This expression can be recognized as:
\[ t = \sinh(x) \]
since \( \sinh(x) = \frac{e^x - e^{-x}}{2} \).

### Conclusion
Thus, we conclude that:
\[ t = \sinh(x) = \frac{e^x - e^{-x}}{2} \]