Simplify ((e^x)/2-(e^(-x))/2)^2

${(\frac{e^{x}}{2} - \frac{e^{- x}}{2})}^{2}$

Step 1

Rewrite ${(\frac{e^{x}}{2} - \frac{e^{- x}}{2})}^{2}$ as $(\frac{e^{x}}{2} - \frac{e^{- x}}{2}) (\frac{e^{x}}{2} - \frac{e^{- x}}{2})$ .

$(\frac{e^{x}}{2} - \frac{e^{- x}}{2}) (\frac{e^{x}}{2} - \frac{e^{- x}}{2})$

Step 2

Expand $(\frac{e^{x}}{2} - \frac{e^{- x}}{2}) (\frac{e^{x}}{2} - \frac{e^{- x}}{2})$ using the FOIL Method.

Apply the distributive property.

$\frac{e^{x}}{2} (\frac{e^{x}}{2} - \frac{e^{- x}}{2}) - \frac{e^{- x}}{2} (\frac{e^{x}}{2} - \frac{e^{- x}}{2})$

Apply the distributive property.

$\frac{e^{x}}{2} \cdot \frac{e^{x}}{2} + \frac{e^{x}}{2} (- \frac{e^{- x}}{2}) - \frac{e^{- x}}{2} (\frac{e^{x}}{2} - \frac{e^{- x}}{2})$

Apply the distributive property.

$\frac{e^{x}}{2} \cdot \frac{e^{x}}{2} + \frac{e^{x}}{2} (- \frac{e^{- x}}{2}) - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Step 3

Simplify and combine like terms.

Simplify each term.

Combine.

$\frac{e^{x} e^{x}}{2 \cdot 2} + \frac{e^{x}}{2} (- \frac{e^{- x}}{2}) - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{e^{x + x}}{2 \cdot 2} + \frac{e^{x}}{2} (- \frac{e^{- x}}{2}) - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Add $x$ and $x$ .

$\frac{e^{2 x}}{2 \cdot 2} + \frac{e^{x}}{2} (- \frac{e^{- x}}{2}) - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Multiply $2$ by $2$ .

$\frac{e^{2 x}}{4} + \frac{e^{x}}{2} (- \frac{e^{- x}}{2}) - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Rewrite using the commutative property of multiplication.

$\frac{e^{2 x}}{4} - \frac{e^{x}}{2} \cdot \frac{e^{- x}}{2} - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Multiply $- \frac{e^{x}}{2} \cdot \frac{e^{- x}}{2}$ .

Multiply $\frac{e^{- x}}{2}$ by $\frac{e^{x}}{2}$ .

$\frac{e^{2 x}}{4} - \frac{e^{- x} e^{x}}{2 \cdot 2} - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Multiply $e^{- x}$ by $e^{x}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{e^{2 x}}{4} - \frac{e^{- x + x}}{2 \cdot 2} - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Add $- x$ and $x$ .

$\frac{e^{2 x}}{4} - \frac{e^{0}}{2 \cdot 2} - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Simplify $e^{0}$ .

$\frac{e^{2 x}}{4} - \frac{1}{2 \cdot 2} - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Multiply $2$ by $2$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Multiply $- \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2}$ .

Multiply $\frac{e^{x}}{2}$ by $\frac{e^{- x}}{2}$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{e^{x} e^{- x}}{2 \cdot 2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Multiply $e^{x}$ by $e^{- x}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{e^{x - x}}{2 \cdot 2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Subtract $x$ from $x$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{e^{0}}{2 \cdot 2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Simplify $e^{0}$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{2 \cdot 2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Multiply $2$ by $2$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{4} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Multiply $- \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$ .

Multiply $- 1$ by $- 1$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{4} + 1 \frac{e^{- x}}{2} \frac{e^{- x}}{2}$

Multiply $\frac{e^{- x}}{2}$ by $1$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{e^{- x}}{2} \cdot \frac{e^{- x}}{2}$

Multiply $\frac{e^{- x}}{2}$ by $\frac{e^{- x}}{2}$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{e^{- x} e^{- x}}{2 \cdot 2}$

Multiply $e^{- x}$ by $e^{- x}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{e^{- x - x}}{2 \cdot 2}$

Subtract $x$ from $- x$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{e^{- 2 x}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{e^{- 2 x}}{4}$

Subtract $\frac{1}{4}$ from $- \frac{1}{4}$ .

$\frac{e^{2 x}}{4} - 2 (\frac{1}{4}) + \frac{e^{- 2 x}}{4}$

Step 4

Simplify each term.

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$\frac{e^{2 x}}{4} + 2 (- 1) \frac{1}{4} + \frac{e^{- 2 x}}{4}$

Factor $2$ out of $4$ .

$\frac{e^{2 x}}{4} + 2 \cdot - 1 \frac{1}{2 \cdot 2} + \frac{e^{- 2 x}}{4}$

Cancel the common factor.

$\frac{e^{2 x}}{4} + 2 \cdot - 1 \frac{1}{2 \cdot 2} + \frac{e^{- 2 x}}{4}$

Rewrite the expression.

$\frac{e^{2 x}}{4} - 1 (\frac{1}{2}) + \frac{e^{- 2 x}}{4}$

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

$\frac{e^{2 x}}{4} - \frac{1}{2} + \frac{e^{- 2 x}}{4}$

Simplify ((e^x)/2-(e^(-x))/2)^2

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