Algebra Examples

$\frac{\tan^{3} (x) - 1}{\tan (x) - 1} = \tan^{2} (x) + \tan (x) + 1$

Step 1

Start on the left side.

$\frac{\tan^{3} (x) - 1}{\tan (x) - 1}$

Step 2

Convert to sines and cosines.

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Step 2.1

Write $\tan (x)$ in sines and cosines using the quotient identity.

$\frac{{(\frac{\sin (x)}{\cos (x)})}^{3} - 1}{\tan (x) - 1}$

Step 2.2

Write $\tan (x)$ in sines and cosines using the quotient identity.

$\frac{{(\frac{\sin (x)}{\cos (x)})}^{3} - 1}{\frac{\sin (x)}{\cos (x)} - 1}$

Step 2.3

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\frac{\sin^{3} (x)}{\cos^{3} (x)} - 1}{\frac{\sin (x)}{\cos (x)} - 1}$

Step 3

Simplify.

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Step 3.1

Multiply the numerator and denominator of the fraction by ${\cos (x)}^{3}$ .

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Step 3.1.1

Multiply $\frac{\frac{{\sin (x)}^{3}}{{\cos (x)}^{3}} - 1}{\frac{\sin (x)}{\cos (x)} - 1}$ by $\frac{{\cos (x)}^{3}}{{\cos (x)}^{3}}$ .

$\frac{{\cos (x)}^{3}}{{\cos (x)}^{3}} \cdot \frac{\frac{{\sin (x)}^{3}}{{\cos (x)}^{3}} - 1}{\frac{\sin (x)}{\cos (x)} - 1}$

Step 3.1.2

Combine.

$\frac{{\cos (x)}^{3} (\frac{{\sin (x)}^{3}}{{\cos (x)}^{3}} - 1)}{{\cos (x)}^{3} (\frac{\sin (x)}{\cos (x)} - 1)}$

Step 3.2

Apply the distributive property.

$\frac{{\cos (x)}^{3} \frac{{\sin (x)}^{3}}{{\cos (x)}^{3}} + {\cos (x)}^{3} \cdot - 1}{{\cos (x)}^{3} \frac{\sin (x)}{\cos (x)} + {\cos (x)}^{3} \cdot - 1}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of ${\cos (x)}^{3}$ .

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Step 3.3.1.1

Cancel the common factor.

$\frac{{\cos (x)}^{3} \frac{{\sin (x)}^{3}}{{\cos (x)}^{3}} + {\cos (x)}^{3} \cdot - 1}{{\cos (x)}^{3} \frac{\sin (x)}{\cos (x)} + {\cos (x)}^{3} \cdot - 1}$

Step 3.3.1.2

Rewrite the expression.

$\frac{{\sin (x)}^{3} + {\cos (x)}^{3} \cdot - 1}{{\cos (x)}^{3} \frac{\sin (x)}{\cos (x)} + {\cos (x)}^{3} \cdot - 1}$

Step 3.3.2

Cancel the common factor of $\cos (x)$ .

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Step 3.3.2.1

Factor $\cos (x)$ out of ${\cos (x)}^{3}$ .

$\frac{{\sin (x)}^{3} + {\cos (x)}^{3} \cdot - 1}{\cos (x) {\cos (x)}^{2} \frac{\sin (x)}{\cos (x)} + {\cos (x)}^{3} \cdot - 1}$

Step 3.3.2.2

Cancel the common factor.

$\frac{{\sin (x)}^{3} + {\cos (x)}^{3} \cdot - 1}{\cos (x) {\cos (x)}^{2} \frac{\sin (x)}{\cos (x)} + {\cos (x)}^{3} \cdot - 1}$

Step 3.3.2.3

Rewrite the expression.

$\frac{{\sin (x)}^{3} + {\cos (x)}^{3} \cdot - 1}{{\cos (x)}^{2} \sin (x) + {\cos (x)}^{3} \cdot - 1}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Rewrite ${\cos (x)}^{3} \cdot - 1$ as ${(\cos (x) \cdot - 1)}^{3}$ .

$\frac{{\sin (x)}^{3} + {(\cos (x) \cdot - 1)}^{3}}{{\cos (x)}^{2} \sin (x) + {\cos (x)}^{3} \cdot - 1}$

Step 3.4.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = \sin (x)$ and $b = \cos (x) \cdot - 1$ .

$\frac{(\sin (x) + \cos (x) \cdot - 1) ({\sin (x)}^{2} - \sin (x) (\cos (x) \cdot - 1) + {(\cos (x) \cdot - 1)}^{2})}{{\cos (x)}^{2} \sin (x) + {\cos (x)}^{3} \cdot - 1}$

Step 3.4.3

Simplify.

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Step 3.4.3.1

Move $- 1$ to the left of $\cos (x)$ .

$\frac{(\sin (x) - 1 \cdot \cos (x)) ({\sin (x)}^{2} - \sin (x) (\cos (x) \cdot - 1) + {(\cos (x) \cdot - 1)}^{2})}{{\cos (x)}^{2} \sin (x) + {\cos (x)}^{3} \cdot - 1}$

Step 3.4.3.2

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

$\frac{(\sin (x) - \cos (x)) ({\sin (x)}^{2} - \sin (x) (\cos (x) \cdot - 1) + {(\cos (x) \cdot - 1)}^{2})}{{\cos (x)}^{2} \sin (x) + {\cos (x)}^{3} \cdot - 1}$

Step 3.4.3.3

Move $- 1$ to the left of $\cos (x)$ .

$\frac{(\sin (x) - \cos (x)) ({\sin (x)}^{2} - \sin (x) (- 1 \cdot \cos (x)) + {(\cos (x) \cdot - 1)}^{2})}{{\cos (x)}^{2} \sin (x) + {\cos (x)}^{3} \cdot - 1}$

Step 3.4.3.4

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

$\frac{(\sin (x) - \cos (x)) ({\sin (x)}^{2} - \sin (x) (- \cos (x)) + {(\cos (x) \cdot - 1)}^{2})}{{\cos (x)}^{2} \sin (x) + {\cos (x)}^{3} \cdot - 1}$

Step 3.4.3.5

Multiply $- \sin (x) (- \cos (x))$ .

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Step 3.4.3.5.1

Multiply $- 1$ by $- 1$ .

$\frac{(\sin (x) - \cos (x)) ({\sin (x)}^{2} + 1 \sin (x) \cos (x) + {(\cos (x) \cdot - 1)}^{2})}{{\cos (x)}^{2} \sin (x) + {\cos (x)}^{3} \cdot - 1}$

Step 3.4.3.5.2

Multiply $\sin (x)$ by $1$ .

$\frac{(\sin (x) - \cos (x)) ({\sin (x)}^{2} + \sin (x) \cos (x) + {(\cos (x) \cdot - 1)}^{2})}{{\cos (x)}^{2} \sin (x) + {\cos (x)}^{3} \cdot - 1}$

Step 3.4.3.6

Move $- 1$ to the left of $\cos (x)$ .

$\frac{(\sin (x) - \cos (x)) ({\sin (x)}^{2} + \sin (x) \cos (x) + {(- 1 \cdot \cos (x))}^{2})}{{\cos (x)}^{2} \sin (x) + {\cos (x)}^{3} \cdot - 1}$

Step 3.4.3.7

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

$\frac{(\sin (x) - \cos (x)) ({\sin (x)}^{2} + \sin (x) \cos (x) + {(- \cos (x))}^{2})}{{\cos (x)}^{2} \sin (x) + {\cos (x)}^{3} \cdot - 1}$

Step 3.4.3.8

Apply the product rule to $- \cos (x)$ .

$\frac{(\sin (x) - \cos (x)) ({\sin (x)}^{2} + \sin (x) \cos (x) + {(- 1)}^{2} {\cos (x)}^{2})}{{\cos (x)}^{2} \sin (x) + {\cos (x)}^{3} \cdot - 1}$

Step 3.4.3.9

Raise $- 1$ to the power of $2$ .

$\frac{(\sin (x) - \cos (x)) ({\sin (x)}^{2} + \sin (x) \cos (x) + 1 {\cos (x)}^{2})}{{\cos (x)}^{2} \sin (x) + {\cos (x)}^{3} \cdot - 1}$

Step 3.4.3.10

Multiply ${\cos (x)}^{2}$ by $1$ .

$\frac{(\sin (x) - \cos (x)) ({\sin (x)}^{2} + \sin (x) \cos (x) + {\cos (x)}^{2})}{{\cos (x)}^{2} \sin (x) + {\cos (x)}^{3} \cdot - 1}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Factor ${\cos (x)}^{2}$ out of ${\cos (x)}^{2} \sin (x) + {\cos (x)}^{3} \cdot - 1$ .

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Step 3.5.1.1

Factor ${\cos (x)}^{2}$ out of ${\cos (x)}^{2} \sin (x)$ .

$\frac{(\sin (x) - \cos (x)) ({\sin (x)}^{2} + \sin (x) \cos (x) + {\cos (x)}^{2})}{{\cos (x)}^{2} (\sin (x)) + {\cos (x)}^{3} \cdot - 1}$

Step 3.5.1.2

Factor ${\cos (x)}^{2}$ out of ${\cos (x)}^{3} \cdot - 1$ .

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

Step 3.5.1.3

Factor ${\cos (x)}^{2}$ out of ${\cos (x)}^{2} (\sin (x)) + {\cos (x)}^{2} (\cos (x) \cdot - 1)$ .

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

Step 3.5.2

Move $- 1$ to the left of $\cos (x)$ .

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

Step 3.5.3

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

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

Step 3.6

Cancel the common factor of $\sin (x) - \cos (x)$ .

$\frac{\sin^{2} (x) + \sin (x) \cos (x) + \cos^{2} (x)}{\cos^{2} (x)}$

Step 4

Reorder terms.

$\frac{\cos^{2} (x) + \sin^{2} (x) + \cos (x) \sin (x)}{\cos^{2} (x)}$

Step 5

Rewrite $\frac{\cos^{2} (x) + \sin^{2} (x) + \cos (x) \sin (x)}{\cos^{2} (x)}$ as $\tan^{2} (x) + \tan (x) + 1$ .

$\tan^{2} (x) + \tan (x) + 1$

Step 6

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{\tan^{3} (x) - 1}{\tan (x) - 1} = \tan^{2} (x) + \tan (x) + 1$ is an identity