The value of `((e^x + e^(-x))/(2))^(2) - ((e^x - e^(-x))/(2))^(2)` is :

Options

A. 0
B. 1
C. 4
D. 16

Answer

To solve the expression \(\left(\frac{e^x + e^{-x}}{2}\right)^{2} - \left(\frac{e^x - e^{-x}}{2}\right)^{2}\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \left(\frac{e^x + e^{-x}}{2}\right)^{2} - \left(\frac{e^x - e^{-x}}{2}\right)^{2} \]

### Step 2: Factor out the common term
Notice that both terms have a common factor of \(\frac{1}{4}\):
\[ \frac{1}{4} \left( (e^x + e^{-x})^{2} - (e^x - e^{-x})^{2} \right) \]

### Step 3: Apply the difference of squares formula
The expression inside the parentheses is a difference of squares, which can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\). Let \(a = e^x + e^{-x}\) and \(b = e^x - e^{-x}\):
\[ = \frac{1}{4} \left( (e^x + e^{-x} - (e^x - e^{-x}))(e^x + e^{-x} + (e^x - e^{-x})) \right) \]

### Step 4: Simplify the factors
Now simplify the factors:
1. \(e^x + e^{-x} - (e^x - e^{-x}) = e^{-x} + e^{-x} = 2e^{-x}\)
2. \(e^x + e^{-x} + (e^x - e^{-x}) = e^x + e^x = 2e^x\)

Putting it all together:
\[ = \frac{1}{4} \left( 2e^{-x} \cdot 2e^x \right) \]

### Step 5: Simplify further
Now simplify the expression:
\[ = \frac{1}{4} \cdot 4 = 1 \]

### Final Answer
Thus, the value of the original expression is:
\[ \boxed{1} \]

---