Trigonometry Examples

$(x + \frac{1}{x}) (x - \frac{1}{x})$

Step 1

Expand $(x + \frac{1}{x}) (x - \frac{1}{x})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$x (x - \frac{1}{x}) + \frac{1}{x} (x - \frac{1}{x})$

Step 1.2

Apply the distributive property.

$x \cdot x + x (- \frac{1}{x}) + \frac{1}{x} (x - \frac{1}{x})$

Step 1.3

Apply the distributive property.

$x \cdot x + x (- \frac{1}{x}) + \frac{1}{x} x + \frac{1}{x} (- \frac{1}{x})$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $x \cdot x + x (- \frac{1}{x}) + \frac{1}{x} x + \frac{1}{x} (- \frac{1}{x})$ .

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Step 2.1.1

Reorder the factors in the terms $x (- \frac{1}{x})$ and $\frac{1}{x} x$ .

$x \cdot x - x \frac{1}{x} + x \frac{1}{x} + \frac{1}{x} (- \frac{1}{x})$

Step 2.1.2

Add $- x \frac{1}{x}$ and $x \frac{1}{x}$ .

$x \cdot x + 0 + \frac{1}{x} (- \frac{1}{x})$

Step 2.1.3

Add $x \cdot x$ and $0$ .

$x \cdot x + \frac{1}{x} (- \frac{1}{x})$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $x$ by $x$ .

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

Step 2.2.2

Rewrite using the commutative property of multiplication.

$x^{2} - \frac{1}{x} \cdot \frac{1}{x}$

Step 2.2.3

Multiply $- \frac{1}{x} \cdot \frac{1}{x}$ .

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Step 2.2.3.1

Multiply $\frac{1}{x}$ by $\frac{1}{x}$ .

$x^{2} - \frac{1}{x \cdot x}$

Step 2.2.3.2

Raise $x$ to the power of $1$ .

$x^{2} - \frac{1}{x^{1} x}$

Step 2.2.3.3

Raise $x$ to the power of $1$ .

$x^{2} - \frac{1}{x^{1} x^{1}}$

Step 2.2.3.4

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

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

Step 2.2.3.5

Add $1$ and $1$ .

$x^{2} - \frac{1}{x^{2}}$