Algebra Examples

$\tan (x - \frac{π}{3})$

Step 1

Apply the difference of angles identity.

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

Step 2

The exact value of $\tan (\frac{π}{3})$ is $\sqrt{3}$ .

$\frac{\tan (x) - \sqrt{3}}{1 + \tan (x) \tan (\frac{π}{3})}$

Step 3

The exact value of $\tan (\frac{π}{3})$ is $\sqrt{3}$ .

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

Step 4

Multiply $\frac{\tan (x) - \sqrt{3}}{1 + \tan (x) \sqrt{3}}$ by $\frac{1 - \tan (x) \sqrt{3}}{1 - \tan (x) \sqrt{3}}$ .

$\frac{\tan (x) - \sqrt{3}}{1 + \tan (x) \sqrt{3}} \cdot \frac{1 - \tan (x) \sqrt{3}}{1 - \tan (x) \sqrt{3}}$

Step 5

Combine fractions.

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Step 5.1

Multiply $\frac{\tan (x) - \sqrt{3}}{1 + \tan (x) \sqrt{3}}$ by $\frac{1 - \tan (x) \sqrt{3}}{1 - \tan (x) \sqrt{3}}$ .

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

Step 5.2

Expand the denominator using the FOIL method.

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

Step 5.3

Simplify.

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Step 5.3.1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 5.3.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{(\tan (x) - \sqrt{3}) (1 - \tan (x) \sqrt{3})}{1 - \tan^{2} (x) \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 5.3.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{(\tan (x) - \sqrt{3}) (1 - \tan (x) \sqrt{3})}{1 - \tan^{2} (x) \cdot 3^{\frac{2}{2}}}$

Step 5.3.1.4

Cancel the common factor of $2$ .

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Step 5.3.1.4.1

Cancel the common factor.

$\frac{(\tan (x) - \sqrt{3}) (1 - \tan (x) \sqrt{3})}{1 - \tan^{2} (x) \cdot 3^{\frac{2}{2}}}$

Step 5.3.1.4.2

Rewrite the expression.

$\frac{(\tan (x) - \sqrt{3}) (1 - \tan (x) \sqrt{3})}{1 - \tan^{2} (x) \cdot 3^{1}}$

Step 5.3.1.5

Evaluate the exponent.

$\frac{(\tan (x) - \sqrt{3}) (1 - \tan (x) \sqrt{3})}{1 - \tan^{2} (x) \cdot 3}$

Step 5.3.2

Multiply $3$ by $- 1$ .

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

Step 6

Expand $(\tan (x) - \sqrt{3}) (1 - \tan (x) \sqrt{3})$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

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

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

Step 7.1.2

Rewrite using the commutative property of multiplication.

$\frac{\tan (x) - \sqrt{3} \tan (x) \tan (x) - \sqrt{3} \cdot 1 - \sqrt{3} (- \tan (x) \sqrt{3})}{1 - 3 \tan^{2} (x)}$

Step 7.1.3

Multiply $- \sqrt{3} \tan (x) \tan (x)$ .

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Step 7.1.3.1

Raise $\tan (x)$ to the power of $1$ .

$\frac{\tan (x) - \sqrt{3} (\tan^{1} (x) \tan (x)) - \sqrt{3} \cdot 1 - \sqrt{3} (- \tan (x) \sqrt{3})}{1 - 3 \tan^{2} (x)}$

Step 7.1.3.2

Raise $\tan (x)$ to the power of $1$ .

$\frac{\tan (x) - \sqrt{3} (\tan^{1} (x) \tan^{1} (x)) - \sqrt{3} \cdot 1 - \sqrt{3} (- \tan (x) \sqrt{3})}{1 - 3 \tan^{2} (x)}$

Step 7.1.3.3

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

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

Step 7.1.3.4

Add $1$ and $1$ .

$\frac{\tan (x) - \sqrt{3} \tan^{2} (x) - \sqrt{3} \cdot 1 - \sqrt{3} (- \tan (x) \sqrt{3})}{1 - 3 \tan^{2} (x)}$

Step 7.1.4

Multiply $- 1$ by $1$ .

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

Step 7.1.5

Multiply $- \sqrt{3} (- \tan (x) \sqrt{3})$ .

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Step 7.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 7.1.5.2

Multiply $\sqrt{3}$ by $1$ .

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

Step 7.1.5.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 7.1.5.4

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 7.1.5.5

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

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

Step 7.1.5.6

Add $1$ and $1$ .

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

Step 7.1.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 7.1.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 7.1.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 7.1.6.4

Cancel the common factor of $2$ .

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Step 7.1.6.4.1

Cancel the common factor.

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

Step 7.1.6.4.2

Rewrite the expression.

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

Step 7.1.6.5

Evaluate the exponent.

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

Step 7.2

Add $\tan (x)$ and $3 \tan (x)$ .

$\frac{4 \tan (x) - \sqrt{3} \tan^{2} (x) - \sqrt{3}}{1 - 3 \tan^{2} (x)}$