Trigonometry Examples

$\csc^{2} (75) - \tan^{2} (15)$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\csc (75)$ is $- \sqrt{2} + \sqrt{6}$ .

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Step 1.1.1

Split $75$ into two angles where the values of the six trigonometric functions are known.

$\csc^{2} (30 + 45) - \tan^{2} (15)$

Step 1.1.2

Apply the sum of angles identity.

${(\frac{\sec (30) \sec (45) \csc (30) \csc (45)}{\sec (30) \csc (45) + \csc (30) \sec (45)})}^{2} - \tan^{2} (15)$

Step 1.1.3

The exact value of $\sec (30)$ is $\frac{2}{\sqrt{3}}$ .

${(\frac{\frac{2}{\sqrt{3}} \sec (45) \csc (30) \csc (45)}{\sec (30) \csc (45) + \csc (30) \sec (45)})}^{2} - \tan^{2} (15)$

Step 1.1.4

The exact value of $\sec (45)$ is $\frac{2}{\sqrt{2}}$ .

${(\frac{\frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{2}} \csc (30) \csc (45)}{\sec (30) \csc (45) + \csc (30) \sec (45)})}^{2} - \tan^{2} (15)$

Step 1.1.5

The exact value of $\csc (30)$ is $2$ .

${(\frac{\frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{2}} \cdot 2 \csc (45)}{\sec (30) \csc (45) + \csc (30) \sec (45)})}^{2} - \tan^{2} (15)$

Step 1.1.6

The exact value of $\csc (45)$ is $\sqrt{2}$ .

${(\frac{\frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{2}} \cdot 2 \sqrt{2}}{\sec (30) \csc (45) + \csc (30) \sec (45)})}^{2} - \tan^{2} (15)$

Step 1.1.7

The exact value of $\sec (30)$ is $\frac{2}{\sqrt{3}}$ .

${(\frac{\frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{2}} \cdot 2 \sqrt{2}}{\frac{2}{\sqrt{3}} \csc (45) + \csc (30) \sec (45)})}^{2} - \tan^{2} (15)$

Step 1.1.8

The exact value of $\csc (45)$ is $\sqrt{2}$ .

${(\frac{\frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{2}} \cdot 2 \sqrt{2}}{\frac{2}{\sqrt{3}} \sqrt{2} + \csc (30) \sec (45)})}^{2} - \tan^{2} (15)$

Step 1.1.9

The exact value of $\csc (30)$ is $2$ .

${(\frac{\frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{2}} \cdot 2 \sqrt{2}}{\frac{2}{\sqrt{3}} \sqrt{2} + 2 \sec (45)})}^{2} - \tan^{2} (15)$

Step 1.1.10

The exact value of $\sec (45)$ is $\frac{2}{\sqrt{2}}$ .

${(\frac{\frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{2}} \cdot 2 \sqrt{2}}{\frac{2}{\sqrt{3}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11

Simplify $\frac{\frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{2}} \cdot 2 \sqrt{2}}{\frac{2}{\sqrt{3}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$ .

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Step 1.1.11.1

Simplify the numerator.

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Step 1.1.11.1.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{2}{\sqrt{2}}$ .

${(\frac{\frac{2 \cdot 2}{\sqrt{3} \sqrt{2}} \cdot 2 \sqrt{2}}{\frac{2}{\sqrt{3}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.1.2

Combine $\frac{2 \cdot 2}{\sqrt{3} \sqrt{2}}$ and $2$ .

${(\frac{\frac{2 \cdot 2 \cdot 2}{\sqrt{3} \sqrt{2}} \sqrt{2}}{\frac{2}{\sqrt{3}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.1.3

Combine $\frac{2 \cdot 2 \cdot 2}{\sqrt{3} \sqrt{2}}$ and $\sqrt{2}$ .

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2}{\sqrt{3}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2

Simplify the denominator.

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Step 1.1.11.2.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.2

Combine and simplify the denominator.

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Step 1.1.11.2.2.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.2.2

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

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.2.3

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

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.2.4

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

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.2.5

Add $1$ and $1$ .

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.2.6

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

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Step 1.1.11.2.2.6.1

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

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.2.6.2

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

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.2.6.3

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

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.2.6.4

Cancel the common factor of $2$ .

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Step 1.1.11.2.2.6.4.1

Cancel the common factor.

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.2.6.4.2

Rewrite the expression.

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{3^{1}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.2.6.5

Evaluate the exponent.

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{3} \sqrt{2} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.3

Multiply $\frac{2 \sqrt{3}}{3} \sqrt{2}$ .

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Step 1.1.11.2.3.1

Combine $\frac{2 \sqrt{3}}{3}$ and $\sqrt{2}$ .

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3} \sqrt{2}}{3} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.3.2

Combine using the product rule for radicals.

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{2 \cdot 3}}{3} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.3.3

Multiply $2$ by $3$ .

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2}{\sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.4

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 (\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.5

Combine and simplify the denominator.

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Step 1.1.11.2.5.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.5.2

Raise $\sqrt{2}$ to the power of $1$ .

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.5.3

Raise $\sqrt{2}$ to the power of $1$ .

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.5.4

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

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.5.5

Add $1$ and $1$ .

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.5.6

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

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Step 1.1.11.2.5.6.1

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

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.5.6.2

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

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.5.6.3

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

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{2^{\frac{2}{2}}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.5.6.4

Cancel the common factor of $2$ .

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Step 1.1.11.2.5.6.4.1

Cancel the common factor.

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{2^{\frac{2}{2}}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.5.6.4.2

Rewrite the expression.

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{2^{1}}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.5.6.5

Evaluate the exponent.

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{2}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.6

Cancel the common factor of $2$ .

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Step 1.1.11.2.6.1

Cancel the common factor.

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{2}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.6.2

Rewrite the expression.

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \sqrt{2}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.7

To write $2 \sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \sqrt{2} \cdot \frac{3}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.8

Combine $2 \sqrt{2}$ and $\frac{3}{3}$ .

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + \frac{2 \sqrt{2} \cdot 3}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.9

Combine the numerators over the common denominator.

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6} + 2 \sqrt{2} \cdot 3}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.2.10

Multiply $3$ by $2$ .

${(\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.3

Simplify the numerator.

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Step 1.1.11.3.1

Multiply $2$ by $2$ .

${(\frac{\frac{4 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.3.2

Multiply $4$ by $2$ .

${(\frac{\frac{8 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.4

Simplify the denominator.

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Step 1.1.11.4.1

Combine using the product rule for radicals.

${(\frac{\frac{8 \sqrt{2}}{\sqrt{3 \cdot 2}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.4.2

Multiply $3$ by $2$ .

${(\frac{\frac{8 \sqrt{2}}{\sqrt{6}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5

Simplify the numerator.

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Step 1.1.11.5.1

Combine $\sqrt{2}$ and $\sqrt{6}$ into a single radical.

${(\frac{8 \sqrt{\frac{2}{6}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.2

Cancel the common factor of $2$ and $6$ .

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Step 1.1.11.5.2.1

Factor $2$ out of $2$ .

${(\frac{8 \sqrt{\frac{2 (1)}{6}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.2.2

Cancel the common factors.

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Step 1.1.11.5.2.2.1

Factor $2$ out of $6$ .

${(\frac{8 \sqrt{\frac{2 \cdot 1}{2 \cdot 3}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.2.2.2

Cancel the common factor.

${(\frac{8 \sqrt{\frac{2 \cdot 1}{2 \cdot 3}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.2.2.3

Rewrite the expression.

${(\frac{8 \sqrt{\frac{1}{3}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.3

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

${(\frac{8 \frac{\sqrt{1}}{\sqrt{3}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.4

Any root of $1$ is $1$ .

${(\frac{8 \frac{1}{\sqrt{3}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.5

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

${(\frac{8 (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.6

Combine and simplify the denominator.

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Step 1.1.11.5.6.1

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

${(\frac{8 \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.6.2

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

${(\frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.6.3

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

${(\frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.6.4

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

${(\frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.6.5

Add $1$ and $1$ .

${(\frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{2}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.6.6

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

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Step 1.1.11.5.6.6.1

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

${(\frac{8 \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.6.6.2

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

${(\frac{8 \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.6.6.3

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

${(\frac{8 \frac{\sqrt{3}}{3^{\frac{2}{2}}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.6.6.4

Cancel the common factor of $2$ .

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Step 1.1.11.5.6.6.4.1

Cancel the common factor.

${(\frac{8 \frac{\sqrt{3}}{3^{\frac{2}{2}}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.6.6.4.2

Rewrite the expression.

${(\frac{8 \frac{\sqrt{3}}{3^{1}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.6.6.5

Evaluate the exponent.

${(\frac{8 \frac{\sqrt{3}}{3}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.5.7

Combine $8$ and $\frac{\sqrt{3}}{3}$ .

${(\frac{\frac{8 \sqrt{3}}{3}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}})}^{2} - \tan^{2} (15)$

Step 1.1.11.6

Multiply the numerator by the reciprocal of the denominator.

${(\frac{8 \sqrt{3}}{3} \cdot \frac{3}{2 \sqrt{6} + 6 \sqrt{2}})}^{2} - \tan^{2} (15)$

Step 1.1.11.7

Cancel the common factor of $3$ .

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Step 1.1.11.7.1

Cancel the common factor.

${(\frac{8 \sqrt{3}}{3} \cdot \frac{3}{2 \sqrt{6} + 6 \sqrt{2}})}^{2} - \tan^{2} (15)$

Step 1.1.11.7.2

Rewrite the expression.

${(8 \sqrt{3} \frac{1}{2 \sqrt{6} + 6 \sqrt{2}})}^{2} - \tan^{2} (15)$

Step 1.1.11.8

Combine $\frac{1}{2 \sqrt{6} + 6 \sqrt{2}}$ and $8$ .

${(\frac{8}{2 \sqrt{6} + 6 \sqrt{2}} \sqrt{3})}^{2} - \tan^{2} (15)$

Step 1.1.11.9

Combine $\frac{8}{2 \sqrt{6} + 6 \sqrt{2}}$ and $\sqrt{3}$ .

${(\frac{8 \sqrt{3}}{2 \sqrt{6} + 6 \sqrt{2}})}^{2} - \tan^{2} (15)$

Step 1.1.11.10

Cancel the common factor of $8$ and $2 \sqrt{6} + 6 \sqrt{2}$ .

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Step 1.1.11.10.1

Factor $2$ out of $8 \sqrt{3}$ .

${(\frac{2 (4 \sqrt{3})}{2 \sqrt{6} + 6 \sqrt{2}})}^{2} - \tan^{2} (15)$

Step 1.1.11.10.2

Cancel the common factors.

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Step 1.1.11.10.2.1

Factor $2$ out of $2 \sqrt{6}$ .

${(\frac{2 (4 \sqrt{3})}{2 (\sqrt{6}) + 6 \sqrt{2}})}^{2} - \tan^{2} (15)$

Step 1.1.11.10.2.2

Factor $2$ out of $6 \sqrt{2}$ .

${(\frac{2 (4 \sqrt{3})}{2 (\sqrt{6}) + 2 (3 \sqrt{2})})}^{2} - \tan^{2} (15)$

Step 1.1.11.10.2.3

Factor $2$ out of $2 (\sqrt{6}) + 2 (3 \sqrt{2})$ .

${(\frac{2 (4 \sqrt{3})}{2 (\sqrt{6} + 3 \sqrt{2})})}^{2} - \tan^{2} (15)$

Step 1.1.11.10.2.4

Cancel the common factor.

${(\frac{2 (4 \sqrt{3})}{2 (\sqrt{6} + 3 \sqrt{2})})}^{2} - \tan^{2} (15)$

Step 1.1.11.10.2.5

Rewrite the expression.

${(\frac{4 \sqrt{3}}{\sqrt{6} + 3 \sqrt{2}})}^{2} - \tan^{2} (15)$

Step 1.1.11.11

Multiply $\frac{4 \sqrt{3}}{\sqrt{6} + 3 \sqrt{2}}$ by $\frac{\sqrt{6} - 3 \sqrt{2}}{\sqrt{6} - 3 \sqrt{2}}$ .

${(\frac{4 \sqrt{3}}{\sqrt{6} + 3 \sqrt{2}} \cdot \frac{\sqrt{6} - 3 \sqrt{2}}{\sqrt{6} - 3 \sqrt{2}})}^{2} - \tan^{2} (15)$

Step 1.1.11.12

Multiply $\frac{4 \sqrt{3}}{\sqrt{6} + 3 \sqrt{2}}$ by $\frac{\sqrt{6} - 3 \sqrt{2}}{\sqrt{6} - 3 \sqrt{2}}$ .

${(\frac{4 \sqrt{3} (\sqrt{6} - 3 \sqrt{2})}{(\sqrt{6} + 3 \sqrt{2}) (\sqrt{6} - 3 \sqrt{2})})}^{2} - \tan^{2} (15)$

Step 1.1.11.13

Expand the denominator using the FOIL method.

${(\frac{4 \sqrt{3} (\sqrt{6} - 3 \sqrt{2})}{{\sqrt{6}}^{2} - 3 \sqrt{12} + 3 \sqrt{12} - 9 {\sqrt{2}}^{2}})}^{2} - \tan^{2} (15)$

Step 1.1.11.14

Simplify.

${(\frac{4 \sqrt{3} (\sqrt{6} - 3 \sqrt{2})}{- 12})}^{2} - \tan^{2} (15)$

Step 1.1.11.15

Cancel the common factor of $4$ and $- 12$ .

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Step 1.1.11.15.1

Factor $4$ out of $4 \sqrt{3} (\sqrt{6} - 3 \sqrt{2})$ .

${(\frac{4 (\sqrt{3} (\sqrt{6} - 3 \sqrt{2}))}{- 12})}^{2} - \tan^{2} (15)$

Step 1.1.11.15.2

Cancel the common factors.

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Step 1.1.11.15.2.1

Factor $4$ out of $- 12$ .

${(\frac{4 (\sqrt{3} (\sqrt{6} - 3 \sqrt{2}))}{4 \cdot - 3})}^{2} - \tan^{2} (15)$

Step 1.1.11.15.2.2

Cancel the common factor.

${(\frac{4 (\sqrt{3} (\sqrt{6} - 3 \sqrt{2}))}{4 \cdot - 3})}^{2} - \tan^{2} (15)$

Step 1.1.11.15.2.3

Rewrite the expression.

${(\frac{\sqrt{3} (\sqrt{6} - 3 \sqrt{2})}{- 3})}^{2} - \tan^{2} (15)$

Step 1.1.11.16

Apply the distributive property.

${(\frac{\sqrt{3} \sqrt{6} + \sqrt{3} (- 3 \sqrt{2})}{- 3})}^{2} - \tan^{2} (15)$

Step 1.1.11.17

Combine using the product rule for radicals.

${(\frac{\sqrt{3 \cdot 6} + \sqrt{3} (- 3 \sqrt{2})}{- 3})}^{2} - \tan^{2} (15)$

Step 1.1.11.18

Multiply $\sqrt{3} (- 3 \sqrt{2})$ .

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Step 1.1.11.18.1

Combine using the product rule for radicals.

${(\frac{\sqrt{3 \cdot 6} - 3 \sqrt{3 \cdot 2}}{- 3})}^{2} - \tan^{2} (15)$

Step 1.1.11.18.2

Multiply $3$ by $2$ .

${(\frac{\sqrt{3 \cdot 6} - 3 \sqrt{6}}{- 3})}^{2} - \tan^{2} (15)$

Step 1.1.11.19

Simplify each term.

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Step 1.1.11.19.1

Multiply $3$ by $6$ .

${(\frac{\sqrt{18} - 3 \sqrt{6}}{- 3})}^{2} - \tan^{2} (15)$

Step 1.1.11.19.2

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 1.1.11.19.2.1

Factor $9$ out of $18$ .

${(\frac{\sqrt{9 (2)} - 3 \sqrt{6}}{- 3})}^{2} - \tan^{2} (15)$

Step 1.1.11.19.2.2

Rewrite $9$ as $3^{2}$ .

${(\frac{\sqrt{3^{2} \cdot 2} - 3 \sqrt{6}}{- 3})}^{2} - \tan^{2} (15)$

Step 1.1.11.19.3

Pull terms out from under the radical.

${(\frac{3 \sqrt{2} - 3 \sqrt{6}}{- 3})}^{2} - \tan^{2} (15)$

Step 1.1.11.20

Cancel the common factor of $3 \sqrt{2} - 3 \sqrt{6}$ and $- 3$ .

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Step 1.1.11.20.1

Factor $3$ out of $3 \sqrt{2}$ .

${(\frac{3 (\sqrt{2}) - 3 \sqrt{6}}{- 3})}^{2} - \tan^{2} (15)$

Step 1.1.11.20.2

Factor $3$ out of $- 3 \sqrt{6}$ .

${(\frac{3 (\sqrt{2}) + 3 (- \sqrt{6})}{- 3})}^{2} - \tan^{2} (15)$

Step 1.1.11.20.3

Factor $3$ out of $3 (\sqrt{2}) + 3 (- \sqrt{6})$ .

${(\frac{3 (\sqrt{2} - \sqrt{6})}{- 3})}^{2} - \tan^{2} (15)$

Step 1.1.11.20.4

Move the negative one from the denominator of $\frac{\sqrt{2} - \sqrt{6}}{- 1}$ .

${(- 1 \cdot (\sqrt{2} - \sqrt{6}))}^{2} - \tan^{2} (15)$

Step 1.1.11.21

Rewrite $- 1 \cdot (\sqrt{2} - \sqrt{6})$ as $- (\sqrt{2} - \sqrt{6})$ .

${(- (\sqrt{2} - \sqrt{6}))}^{2} - \tan^{2} (15)$

Step 1.1.11.22

Apply the distributive property.

${(- \sqrt{2} - - \sqrt{6})}^{2} - \tan^{2} (15)$

Step 1.1.11.23

Multiply $- - \sqrt{6}$ .

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Step 1.1.11.23.1

Multiply $- 1$ by $- 1$ .

${(- \sqrt{2} + 1 \sqrt{6})}^{2} - \tan^{2} (15)$

Step 1.1.11.23.2

Multiply $\sqrt{6}$ by $1$ .

${(- \sqrt{2} + \sqrt{6})}^{2} - \tan^{2} (15)$

Step 1.2

Rewrite ${(- \sqrt{2} + \sqrt{6})}^{2}$ as $(- \sqrt{2} + \sqrt{6}) (- \sqrt{2} + \sqrt{6})$ .

$(- \sqrt{2} + \sqrt{6}) (- \sqrt{2} + \sqrt{6}) - \tan^{2} (15)$

Step 1.3

Expand $(- \sqrt{2} + \sqrt{6}) (- \sqrt{2} + \sqrt{6})$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$- \sqrt{2} (- \sqrt{2} + \sqrt{6}) + \sqrt{6} (- \sqrt{2} + \sqrt{6}) - \tan^{2} (15)$

Step 1.3.2

Apply the distributive property.

$- \sqrt{2} (- \sqrt{2}) - \sqrt{2} \sqrt{6} + \sqrt{6} (- \sqrt{2} + \sqrt{6}) - \tan^{2} (15)$

Step 1.3.3

Apply the distributive property.

$- \sqrt{2} (- \sqrt{2}) - \sqrt{2} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $- \sqrt{2} (- \sqrt{2})$ .

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Step 1.4.1.1.1

Multiply $- 1$ by $- 1$ .

$1 \sqrt{2} \sqrt{2} - \sqrt{2} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.1.2

Multiply $\sqrt{2}$ by $1$ .

$\sqrt{2} \sqrt{2} - \sqrt{2} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.1.3

Raise $\sqrt{2}$ to the power of $1$ .

${\sqrt{2}}^{1} \sqrt{2} - \sqrt{2} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.1.4

Raise $\sqrt{2}$ to the power of $1$ .

${\sqrt{2}}^{1} {\sqrt{2}}^{1} - \sqrt{2} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.1.5

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

${\sqrt{2}}^{1 + 1} - \sqrt{2} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.1.6

Add $1$ and $1$ .

${\sqrt{2}}^{2} - \sqrt{2} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.2

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

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Step 1.4.1.2.1

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

${(2^{\frac{1}{2}})}^{2} - \sqrt{2} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.2.2

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

$2^{\frac{1}{2} \cdot 2} - \sqrt{2} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.2.3

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

$2^{\frac{2}{2}} - \sqrt{2} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.2.4

Cancel the common factor of $2$ .

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Step 1.4.1.2.4.1

Cancel the common factor.

$2^{\frac{2}{2}} - \sqrt{2} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.2.4.2

Rewrite the expression.

$2^{1} - \sqrt{2} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.2.5

Evaluate the exponent.

$2 - \sqrt{2} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.3

Multiply $- \sqrt{2} \sqrt{6}$ .

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Step 1.4.1.3.1

Combine using the product rule for radicals.

$2 - \sqrt{6 \cdot 2} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.3.2

Multiply $6$ by $2$ .

$2 - \sqrt{12} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.4.1.4.1

Factor $4$ out of $12$ .

$2 - \sqrt{4 (3)} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.4.2

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

$2 - \sqrt{2^{2} \cdot 3} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.5

Pull terms out from under the radical.

$2 - (2 \sqrt{3}) + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.6

Multiply $2$ by $- 1$ .

$2 - 2 \sqrt{3} + \sqrt{6} (- \sqrt{2}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.7

Multiply $\sqrt{6} (- \sqrt{2})$ .

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Step 1.4.1.7.1

Combine using the product rule for radicals.

$2 - 2 \sqrt{3} - \sqrt{6 \cdot 2} + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.7.2

Multiply $6$ by $2$ .

$2 - 2 \sqrt{3} - \sqrt{12} + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.8

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.4.1.8.1

Factor $4$ out of $12$ .

$2 - 2 \sqrt{3} - \sqrt{4 (3)} + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.8.2

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

$2 - 2 \sqrt{3} - \sqrt{2^{2} \cdot 3} + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.9

Pull terms out from under the radical.

$2 - 2 \sqrt{3} - (2 \sqrt{3}) + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.10

Multiply $2$ by $- 1$ .

$2 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{6} \sqrt{6} - \tan^{2} (15)$

Step 1.4.1.11

Combine using the product rule for radicals.

$2 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{6 \cdot 6} - \tan^{2} (15)$

Step 1.4.1.12

Multiply $6$ by $6$ .

$2 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{36} - \tan^{2} (15)$

Step 1.4.1.13

Rewrite $36$ as $6^{2}$ .

$2 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{6^{2}} - \tan^{2} (15)$

Step 1.4.1.14

Pull terms out from under the radical, assuming positive real numbers.

$2 - 2 \sqrt{3} - 2 \sqrt{3} + 6 - \tan^{2} (15)$

Step 1.4.2

Add $2$ and $6$ .

$8 - 2 \sqrt{3} - 2 \sqrt{3} - \tan^{2} (15)$

Step 1.4.3

Subtract $2 \sqrt{3}$ from $- 2 \sqrt{3}$ .

$8 - 4 \sqrt{3} - \tan^{2} (15)$

Step 1.5

The exact value of $\tan (15)$ is $2 - \sqrt{3}$ .

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Step 1.5.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$8 - 4 \sqrt{3} - \tan^{2} (45 - 30)$

Step 1.5.2

Separate negation.

$8 - 4 \sqrt{3} - \tan^{2} (45 - (30))$

Step 1.5.3

Apply the difference of angles identity.

$8 - 4 \sqrt{3} - {(\frac{\tan (45) - \tan (30)}{1 + \tan (45) \tan (30)})}^{2}$

Step 1.5.4

The exact value of $\tan (45)$ is $1$ .

$8 - 4 \sqrt{3} - {(\frac{1 - \tan (30)}{1 + \tan (45) \tan (30)})}^{2}$

Step 1.5.5

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$8 - 4 \sqrt{3} - {(\frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (45) \tan (30)})}^{2}$

Step 1.5.6

The exact value of $\tan (45)$ is $1$ .

$8 - 4 \sqrt{3} - {(\frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \tan (30)})}^{2}$

Step 1.5.7

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$8 - 4 \sqrt{3} - {(\frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}})}^{2}$

Step 1.5.8

Simplify $\frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}$ .

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Step 1.5.8.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 1.5.8.1.1

Multiply $\frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}$ by $\frac{3}{3}$ .

$8 - 4 \sqrt{3} - {(\frac{3}{3} \cdot \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}})}^{2}$

Step 1.5.8.1.2

Combine.

$8 - 4 \sqrt{3} - {(\frac{3 (1 - \frac{\sqrt{3}}{3})}{3 (1 + 1 \frac{\sqrt{3}}{3})})}^{2}$

Step 1.5.8.2

Apply the distributive property.

$8 - 4 \sqrt{3} - {(\frac{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3})}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})})}^{2}$

Step 1.5.8.3

Cancel the common factor of $3$ .

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Step 1.5.8.3.1

Move the leading negative in $- \frac{\sqrt{3}}{3}$ into the numerator.

$8 - 4 \sqrt{3} - {(\frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})})}^{2}$

Step 1.5.8.3.2

Cancel the common factor.

$8 - 4 \sqrt{3} - {(\frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})})}^{2}$

Step 1.5.8.3.3

Rewrite the expression.

$8 - 4 \sqrt{3} - {(\frac{3 \cdot 1 - \sqrt{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})})}^{2}$

Step 1.5.8.4

Multiply $3$ by $1$ .

$8 - 4 \sqrt{3} - {(\frac{3 - \sqrt{3}}{3 \cdot 1 + 3 \cdot 1 \frac{\sqrt{3}}{3}})}^{2}$

Step 1.5.8.5

Simplify the denominator.

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Step 1.5.8.5.1

Multiply $3$ by $1$ .

$8 - 4 \sqrt{3} - {(\frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}})}^{2}$

Step 1.5.8.5.2

Cancel the common factor of $3$ .

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Step 1.5.8.5.2.1

Factor $3$ out of $3 \cdot 1$ .

$8 - 4 \sqrt{3} - {(\frac{3 - \sqrt{3}}{3 + 3 (1) \frac{\sqrt{3}}{3}})}^{2}$

Step 1.5.8.5.2.2

Cancel the common factor.

$8 - 4 \sqrt{3} - {(\frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}})}^{2}$

Step 1.5.8.5.2.3

Rewrite the expression.

$8 - 4 \sqrt{3} - {(\frac{3 - \sqrt{3}}{3 + \sqrt{3}})}^{2}$

Step 1.5.8.6

Multiply $\frac{3 - \sqrt{3}}{3 + \sqrt{3}}$ by $\frac{3 - \sqrt{3}}{3 - \sqrt{3}}$ .

$8 - 4 \sqrt{3} - {(\frac{3 - \sqrt{3}}{3 + \sqrt{3}} \cdot \frac{3 - \sqrt{3}}{3 - \sqrt{3}})}^{2}$

Step 1.5.8.7

Multiply $\frac{3 - \sqrt{3}}{3 + \sqrt{3}}$ by $\frac{3 - \sqrt{3}}{3 - \sqrt{3}}$ .

$8 - 4 \sqrt{3} - {(\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{(3 + \sqrt{3}) (3 - \sqrt{3})})}^{2}$

Step 1.5.8.8

Expand the denominator using the FOIL method.

$8 - 4 \sqrt{3} - {(\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}})}^{2}$

Step 1.5.8.9

Simplify.

$8 - 4 \sqrt{3} - {(\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6})}^{2}$

Step 1.5.8.10

Simplify the numerator.

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Step 1.5.8.10.1

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

$8 - 4 \sqrt{3} - {(\frac{{(3 - \sqrt{3})}^{1} (3 - \sqrt{3})}{6})}^{2}$

Step 1.5.8.10.2

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

$8 - 4 \sqrt{3} - {(\frac{{(3 - \sqrt{3})}^{1} {(3 - \sqrt{3})}^{1}}{6})}^{2}$

Step 1.5.8.10.3

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

$8 - 4 \sqrt{3} - {(\frac{{(3 - \sqrt{3})}^{1 + 1}}{6})}^{2}$

Step 1.5.8.10.4

Add $1$ and $1$ .

$8 - 4 \sqrt{3} - {(\frac{{(3 - \sqrt{3})}^{2}}{6})}^{2}$

Step 1.5.8.11

Rewrite ${(3 - \sqrt{3})}^{2}$ as $(3 - \sqrt{3}) (3 - \sqrt{3})$ .

$8 - 4 \sqrt{3} - {(\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6})}^{2}$

Step 1.5.8.12

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

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Step 1.5.8.12.1

Apply the distributive property.

$8 - 4 \sqrt{3} - {(\frac{3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6})}^{2}$

Step 1.5.8.12.2

Apply the distributive property.

$8 - 4 \sqrt{3} - {(\frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6})}^{2}$

Step 1.5.8.12.3

Apply the distributive property.

$8 - 4 \sqrt{3} - {(\frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6})}^{2}$

Step 1.5.8.13

Simplify and combine like terms.

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Step 1.5.8.13.1

Simplify each term.

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Step 1.5.8.13.1.1

Multiply $3$ by $3$ .

$8 - 4 \sqrt{3} - {(\frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6})}^{2}$

Step 1.5.8.13.1.2

Multiply $- 1$ by $3$ .

$8 - 4 \sqrt{3} - {(\frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6})}^{2}$

Step 1.5.8.13.1.3

Multiply $3$ by $- 1$ .

$8 - 4 \sqrt{3} - {(\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6})}^{2}$

Step 1.5.8.13.1.4

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

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Step 1.5.8.13.1.4.1

Multiply $- 1$ by $- 1$ .

$8 - 4 \sqrt{3} - {(\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6})}^{2}$

Step 1.5.8.13.1.4.2

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

$8 - 4 \sqrt{3} - {(\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6})}^{2}$

Step 1.5.8.13.1.4.3

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

$8 - 4 \sqrt{3} - {(\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6})}^{2}$

Step 1.5.8.13.1.4.4

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

$8 - 4 \sqrt{3} - {(\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6})}^{2}$

Step 1.5.8.13.1.4.5

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

$8 - 4 \sqrt{3} - {(\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6})}^{2}$

Step 1.5.8.13.1.4.6

Add $1$ and $1$ .

$8 - 4 \sqrt{3} - {(\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6})}^{2}$

Step 1.5.8.13.1.5

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

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Step 1.5.8.13.1.5.1

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

$8 - 4 \sqrt{3} - {(\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6})}^{2}$

Step 1.5.8.13.1.5.2

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

$8 - 4 \sqrt{3} - {(\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6})}^{2}$

Step 1.5.8.13.1.5.3

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

$8 - 4 \sqrt{3} - {(\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6})}^{2}$

Step 1.5.8.13.1.5.4

Cancel the common factor of $2$ .

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Step 1.5.8.13.1.5.4.1

Cancel the common factor.

$8 - 4 \sqrt{3} - {(\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6})}^{2}$

Step 1.5.8.13.1.5.4.2

Rewrite the expression.

$8 - 4 \sqrt{3} - {(\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6})}^{2}$

Step 1.5.8.13.1.5.5

Evaluate the exponent.

$8 - 4 \sqrt{3} - {(\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6})}^{2}$

Step 1.5.8.13.2

Add $9$ and $3$ .

$8 - 4 \sqrt{3} - {(\frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6})}^{2}$

Step 1.5.8.13.3

Subtract $3 \sqrt{3}$ from $- 3 \sqrt{3}$ .

$8 - 4 \sqrt{3} - {(\frac{12 - 6 \sqrt{3}}{6})}^{2}$

Step 1.5.8.14

Cancel the common factor of $12 - 6 \sqrt{3}$ and $6$ .

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Step 1.5.8.14.1

Factor $6$ out of $12$ .

$8 - 4 \sqrt{3} - {(\frac{6 \cdot 2 - 6 \sqrt{3}}{6})}^{2}$

Step 1.5.8.14.2

Factor $6$ out of $- 6 \sqrt{3}$ .

$8 - 4 \sqrt{3} - {(\frac{6 \cdot 2 + 6 (- \sqrt{3})}{6})}^{2}$

Step 1.5.8.14.3

Factor $6$ out of $6 (2) + 6 (- \sqrt{3})$ .

$8 - 4 \sqrt{3} - {(\frac{6 (2 - \sqrt{3})}{6})}^{2}$

Step 1.5.8.14.4

Cancel the common factors.

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Step 1.5.8.14.4.1

Factor $6$ out of $6$ .

$8 - 4 \sqrt{3} - {(\frac{6 (2 - \sqrt{3})}{6 (1)})}^{2}$

Step 1.5.8.14.4.2

Cancel the common factor.

$8 - 4 \sqrt{3} - {(\frac{6 (2 - \sqrt{3})}{6 \cdot 1})}^{2}$

Step 1.5.8.14.4.3

Rewrite the expression.

$8 - 4 \sqrt{3} - {(\frac{2 - \sqrt{3}}{1})}^{2}$

Step 1.5.8.14.4.4

Divide $2 - \sqrt{3}$ by $1$ .

$8 - 4 \sqrt{3} - {(2 - \sqrt{3})}^{2}$

Step 1.6

Rewrite ${(2 - \sqrt{3})}^{2}$ as $(2 - \sqrt{3}) (2 - \sqrt{3})$ .

$8 - 4 \sqrt{3} - ((2 - \sqrt{3}) (2 - \sqrt{3}))$

Step 1.7

Expand $(2 - \sqrt{3}) (2 - \sqrt{3})$ using the FOIL Method.

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Step 1.7.1

Apply the distributive property.

$8 - 4 \sqrt{3} - (2 (2 - \sqrt{3}) - \sqrt{3} (2 - \sqrt{3}))$

Step 1.7.2

Apply the distributive property.

$8 - 4 \sqrt{3} - (2 \cdot 2 + 2 (- \sqrt{3}) - \sqrt{3} (2 - \sqrt{3}))$

Step 1.7.3

Apply the distributive property.

$8 - 4 \sqrt{3} - (2 \cdot 2 + 2 (- \sqrt{3}) - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{3}))$

Step 1.8

Simplify and combine like terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Multiply $2$ by $2$ .

$8 - 4 \sqrt{3} - (4 + 2 (- \sqrt{3}) - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{3}))$

Step 1.8.1.2

Multiply $- 1$ by $2$ .

$8 - 4 \sqrt{3} - (4 - 2 \sqrt{3} - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{3}))$

Step 1.8.1.3

Multiply $2$ by $- 1$ .

$8 - 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} - \sqrt{3} (- \sqrt{3}))$

Step 1.8.1.4

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

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Step 1.8.1.4.1

Multiply $- 1$ by $- 1$ .

$8 - 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + 1 \sqrt{3} \sqrt{3})$

Step 1.8.1.4.2

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

$8 - 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{3} \sqrt{3})$

Step 1.8.1.4.3

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

$8 - 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3})$

Step 1.8.1.4.4

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

$8 - 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 1.8.1.4.5

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

$8 - 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{1 + 1})$

Step 1.8.1.4.6

Add $1$ and $1$ .

$8 - 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{2})$

Step 1.8.1.5

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

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Step 1.8.1.5.1

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

$8 - 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + {(3^{\frac{1}{2}})}^{2})$

Step 1.8.1.5.2

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

$8 - 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3^{\frac{1}{2} \cdot 2})$

Step 1.8.1.5.3

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

$8 - 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3^{\frac{2}{2}})$

Step 1.8.1.5.4

Cancel the common factor of $2$ .

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Step 1.8.1.5.4.1

Cancel the common factor.

$8 - 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3^{\frac{2}{2}})$

Step 1.8.1.5.4.2

Rewrite the expression.

$8 - 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3^{1})$

Step 1.8.1.5.5

Evaluate the exponent.

$8 - 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3)$

Step 1.8.2

Add $4$ and $3$ .

$8 - 4 \sqrt{3} - (7 - 2 \sqrt{3} - 2 \sqrt{3})$

Step 1.8.3

Subtract $2 \sqrt{3}$ from $- 2 \sqrt{3}$ .

$8 - 4 \sqrt{3} - (7 - 4 \sqrt{3})$

Step 1.9

Apply the distributive property.

$8 - 4 \sqrt{3} - 1 \cdot 7 - (- 4 \sqrt{3})$

Step 1.10

Multiply $- 1$ by $7$ .

$8 - 4 \sqrt{3} - 7 - (- 4 \sqrt{3})$

Step 1.11

Multiply $- 4$ by $- 1$ .

$8 - 4 \sqrt{3} - 7 + 4 \sqrt{3}$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $7$ from $8$ .

$1 - 4 \sqrt{3} + 4 \sqrt{3}$

Step 2.2

Add $- 4 \sqrt{3}$ and $4 \sqrt{3}$ .

$1 + 0$

Step 2.3

Add $1$ and $0$ .

$1$