Trigonometry Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $4 x$ .

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Step 1.1

Multiply $\frac{\frac{x}{4} - \frac{1}{x}}{\frac{1}{2 x} + \frac{1}{4}}$ by $\frac{4 x}{4 x}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $4$ .

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Step 3.1.1

Factor $4$ out of $4 x$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Cancel the common factor of $x$ .

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Step 3.6.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

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

Step 3.6.2

Factor $x$ out of $4 x$ .

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

Step 3.6.3

Cancel the common factor.

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

Step 3.6.4

Rewrite the expression.

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

Step 3.7

Multiply $4$ by $- 1$ .

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

Step 3.8

Cancel the common factor of $2 x$ .

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Step 3.8.1

Factor $2 x$ out of $4 x$ .

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

Step 3.8.2

Cancel the common factor.

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

Step 3.8.3

Rewrite the expression.

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

Step 3.9

Cancel the common factor of $4$ .

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Step 3.9.1

Factor $4$ out of $4 x$ .

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

Step 3.9.2

Cancel the common factor.

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

Step 3.9.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $4$ as $2^{2}$ .

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

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Step 5

Cancel the common factor of $x + 2$ and $2 + x$ .

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Step 5.1

Reorder terms.

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Divide $x - 2$ by $1$ .

$x - 2$