Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $\frac{e^{x} + e^{- x}}{2}$ .

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

Step 1.2

Raise $2$ to the power of $2$ .

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

Step 1.3

Apply the product rule to $\frac{e^{x} - e^{- x}}{2}$ .

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

Step 1.4

Raise $2$ to the power of $2$ .

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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Step 3.1

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

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

Step 3.2

Expand $(e^{x} + e^{- x}) (e^{x} + e^{- x})$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Apply the distributive property.

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

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

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

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Step 3.3.1.1.1

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

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

Step 3.3.1.1.2

Add $x$ and $x$ .

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

Step 3.3.1.2

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

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Step 3.3.1.2.1

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

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

Step 3.3.1.2.2

Subtract $x$ from $x$ .

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

Step 3.3.1.3

Simplify $e^{0}$ .

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

Step 3.3.1.4

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

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Step 3.3.1.4.1

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

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

Step 3.3.1.4.2

Add $- x$ and $x$ .

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

Step 3.3.1.5

Simplify $e^{0}$ .

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

Step 3.3.1.6

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

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Step 3.3.1.6.1

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

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

Step 3.3.1.6.2

Subtract $x$ from $- x$ .

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

Step 3.3.2

Add $1$ and $1$ .

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

Step 3.4

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

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

Step 3.5

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

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Step 3.5.1

Apply the distributive property.

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

Step 3.5.2

Apply the distributive property.

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

Step 3.5.3

Apply the distributive property.

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

Step 3.6

Simplify and combine like terms.

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Step 3.6.1

Simplify each term.

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Step 3.6.1.1

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

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Step 3.6.1.1.1

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

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

Step 3.6.1.1.2

Add $x$ and $x$ .

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

Step 3.6.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.6.1.3

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

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Step 3.6.1.3.1

Move $e^{- x}$ .

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

Step 3.6.1.3.2

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

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

Step 3.6.1.3.3

Add $- x$ and $x$ .

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

Step 3.6.1.4

Simplify $- e^{0}$ .

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

Step 3.6.1.5

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

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Step 3.6.1.5.1

Move $e^{x}$ .

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

Step 3.6.1.5.2

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

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

Step 3.6.1.5.3

Subtract $x$ from $x$ .

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

Step 3.6.1.6

Simplify $- e^{0}$ .

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

Step 3.6.1.7

Rewrite using the commutative property of multiplication.

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

Step 3.6.1.8

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

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Step 3.6.1.8.1

Move $e^{- x}$ .

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

Step 3.6.1.8.2

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

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

Step 3.6.1.8.3

Subtract $x$ from $- x$ .

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

Step 3.6.1.9

Multiply $- 1$ by $- 1$ .

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

Step 3.6.1.10

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

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

Step 3.6.2

Subtract $1$ from $- 1$ .

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

Step 3.7

Apply the distributive property.

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

Step 3.8

Multiply $- 1$ by $- 2$ .

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

Step 4

Simplify by adding terms.

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Step 4.1

Combine the opposite terms in $e^{2 x} + 2 + e^{- 2 x} - e^{2 x} + 2 - e^{- 2 x}$ .

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Step 4.1.1

Subtract $e^{2 x}$ from $e^{2 x}$ .

$\frac{0 + 2 + e^{- 2 x} + 2 - e^{- 2 x}}{4}$

Step 4.1.2

Add $0$ and $2$ .

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

Step 4.1.3

Subtract $e^{- 2 x}$ from $e^{- 2 x}$ .

$\frac{2 + 0 + 2}{4}$

Step 4.1.4

Add $2$ and $0$ .

$\frac{2 + 2}{4}$

Step 4.2

Simplify the expression.

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Step 4.2.1

Add $2$ and $2$ .

$\frac{4}{4}$

Step 4.2.2

Divide $4$ by $4$ .

$1$