Basic Math Examples

$8 \sin (\frac{π}{12}) \cos (\frac{π}{12}) - 1$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\sin (\frac{π}{12})$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.1.1

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$8 \sin (\frac{π}{4} - \frac{π}{6}) \cos (\frac{π}{12}) - 1$

Step 1.1.2

Apply the difference of angles identity.

$8 (\sin (\frac{π}{4}) \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})) \cos (\frac{π}{12}) - 1$

Step 1.1.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$8 (\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})) \cos (\frac{π}{12}) - 1$

Step 1.1.4

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$8 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (\frac{π}{4}) \sin (\frac{π}{6})) \cos (\frac{π}{12}) - 1$

Step 1.1.5

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$8 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (\frac{π}{6})) \cos (\frac{π}{12}) - 1$

Step 1.1.6

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$8 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{π}{12}) - 1$

Step 1.1.7

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.1.7.1

Simplify each term.

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Step 1.1.7.1.1

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

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Step 1.1.7.1.1.1

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

$8 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{π}{12}) - 1$

Step 1.1.7.1.1.2

Combine using the product rule for radicals.

$8 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{π}{12}) - 1$

Step 1.1.7.1.1.3

Multiply $2$ by $3$ .

$8 (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{π}{12}) - 1$

Step 1.1.7.1.1.4

Multiply $2$ by $2$ .

$8 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{π}{12}) - 1$

Step 1.1.7.1.2

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.1.7.1.2.1

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

$8 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) \cos (\frac{π}{12}) - 1$

Step 1.1.7.1.2.2

Multiply $2$ by $2$ .

$8 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) \cos (\frac{π}{12}) - 1$

Step 1.1.7.2

Combine the numerators over the common denominator.

$8 \frac{\sqrt{6} - \sqrt{2}}{4} \cos (\frac{π}{12}) - 1$

Step 1.2

Cancel the common factor of $4$ .

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Step 1.2.1

Factor $4$ out of $8$ .

$4 (2) \frac{\sqrt{6} - \sqrt{2}}{4} \cos (\frac{π}{12}) - 1$

Step 1.2.2

Cancel the common factor.

$4 \cdot 2 \frac{\sqrt{6} - \sqrt{2}}{4} \cos (\frac{π}{12}) - 1$

Step 1.2.3

Rewrite the expression.

$2 (\sqrt{6} - \sqrt{2}) \cos (\frac{π}{12}) - 1$

Step 1.3

Apply the distributive property.

$(2 \sqrt{6} + 2 (- \sqrt{2})) \cos (\frac{π}{12}) - 1$

Step 1.4

Multiply $- 1$ by $2$ .

$(2 \sqrt{6} - 2 \sqrt{2}) \cos (\frac{π}{12}) - 1$

Step 1.5

The exact value of $\cos (\frac{π}{12})$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 1.5.1

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$(2 \sqrt{6} - 2 \sqrt{2}) \cos (\frac{π}{4} - \frac{π}{6}) - 1$

Step 1.5.2

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$(2 \sqrt{6} - 2 \sqrt{2}) (\cos (\frac{π}{4}) \cos (\frac{π}{6}) + \sin (\frac{π}{4}) \sin (\frac{π}{6})) - 1$

Step 1.5.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$(2 \sqrt{6} - 2 \sqrt{2}) (\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) + \sin (\frac{π}{4}) \sin (\frac{π}{6})) - 1$

Step 1.5.4

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$(2 \sqrt{6} - 2 \sqrt{2}) (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (\frac{π}{4}) \sin (\frac{π}{6})) - 1$

Step 1.5.5

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$(2 \sqrt{6} - 2 \sqrt{2}) (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (\frac{π}{6})) - 1$

Step 1.5.6

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Step 1.5.7

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.5.7.1

Simplify each term.

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Step 1.5.7.1.1

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

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Step 1.5.7.1.1.1

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

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

Step 1.5.7.1.1.2

Combine using the product rule for radicals.

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

Step 1.5.7.1.1.3

Multiply $2$ by $3$ .

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

Step 1.5.7.1.1.4

Multiply $2$ by $2$ .

$(2 \sqrt{6} - 2 \sqrt{2}) (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - 1$

Step 1.5.7.1.2

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.5.7.1.2.1

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

$(2 \sqrt{6} - 2 \sqrt{2}) (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) - 1$

Step 1.5.7.1.2.2

Multiply $2$ by $2$ .

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

Step 1.5.7.2

Combine the numerators over the common denominator.

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

Step 1.6

Apply the distributive property.

$2 \sqrt{6} \frac{\sqrt{6} + \sqrt{2}}{4} - 2 \sqrt{2} \frac{\sqrt{6} + \sqrt{2}}{4} - 1$

Step 1.7

Cancel the common factor of $2$ .

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Step 1.7.1

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

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

Step 1.7.2

Factor $2$ out of $4$ .

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

Step 1.7.3

Cancel the common factor.

$2 \sqrt{6} \frac{\sqrt{6} + \sqrt{2}}{2 \cdot 2} - 2 \sqrt{2} \frac{\sqrt{6} + \sqrt{2}}{4} - 1$

Step 1.7.4

Rewrite the expression.

$\sqrt{6} \frac{\sqrt{6} + \sqrt{2}}{2} - 2 \sqrt{2} \frac{\sqrt{6} + \sqrt{2}}{4} - 1$

Step 1.8

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

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

Step 1.9

Cancel the common factor of $2$ .

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Step 1.9.1

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

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

Step 1.9.2

Factor $2$ out of $4$ .

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

Step 1.9.3

Cancel the common factor.

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

Step 1.9.4

Rewrite the expression.

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

Step 1.10

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

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

Step 1.11

Combine the numerators over the common denominator.

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

Step 1.12

Simplify each term.

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Step 1.12.1

Apply the distributive property.

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

Step 1.12.2

Combine using the product rule for radicals.

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

Step 1.12.3

Combine using the product rule for radicals.

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

Step 1.12.4

Simplify each term.

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Step 1.12.4.1

Multiply $6$ by $6$ .

$\frac{\sqrt{36} + \sqrt{6 \cdot 2} - (\sqrt{6} + \sqrt{2}) \sqrt{2}}{2} - 1$

Step 1.12.4.2

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

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

Step 1.12.4.3

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

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

Step 1.12.4.4

Multiply $6$ by $2$ .

$\frac{6 + \sqrt{12} - (\sqrt{6} + \sqrt{2}) \sqrt{2}}{2} - 1$

Step 1.12.4.5

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

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Step 1.12.4.5.1

Factor $4$ out of $12$ .

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

Step 1.12.4.5.2

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

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

Step 1.12.4.6

Pull terms out from under the radical.

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

Step 1.12.5

Apply the distributive property.

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

Step 1.12.6

Apply the distributive property.

$\frac{6 + 2 \sqrt{3} - \sqrt{6} \sqrt{2} - \sqrt{2} \sqrt{2}}{2} - 1$

Step 1.12.7

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

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Step 1.12.7.1

Combine using the product rule for radicals.

$\frac{6 + 2 \sqrt{3} - \sqrt{2 \cdot 6} - \sqrt{2} \sqrt{2}}{2} - 1$

Step 1.12.7.2

Multiply $2$ by $6$ .

$\frac{6 + 2 \sqrt{3} - \sqrt{12} - \sqrt{2} \sqrt{2}}{2} - 1$

Step 1.12.8

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

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Step 1.12.8.1

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

$\frac{6 + 2 \sqrt{3} - \sqrt{12} - ({\sqrt{2}}^{1} \sqrt{2})}{2} - 1$

Step 1.12.8.2

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

$\frac{6 + 2 \sqrt{3} - \sqrt{12} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2} - 1$

Step 1.12.8.3

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

$\frac{6 + 2 \sqrt{3} - \sqrt{12} - {\sqrt{2}}^{1 + 1}}{2} - 1$

Step 1.12.8.4

Add $1$ and $1$ .

$\frac{6 + 2 \sqrt{3} - \sqrt{12} - {\sqrt{2}}^{2}}{2} - 1$

Step 1.12.9

Simplify each term.

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Step 1.12.9.1

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

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Step 1.12.9.1.1

Factor $4$ out of $12$ .

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

Step 1.12.9.1.2

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

$\frac{6 + 2 \sqrt{3} - \sqrt{2^{2} \cdot 3} - {\sqrt{2}}^{2}}{2} - 1$

Step 1.12.9.2

Pull terms out from under the radical.

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

Step 1.12.9.3

Multiply $2$ by $- 1$ .

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - {\sqrt{2}}^{2}}{2} - 1$

Step 1.12.9.4

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

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Step 1.12.9.4.1

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

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

Step 1.12.9.4.2

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

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - 2^{\frac{1}{2} \cdot 2}}{2} - 1$

Step 1.12.9.4.3

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

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - 2^{\frac{2}{2}}}{2} - 1$

Step 1.12.9.4.4

Cancel the common factor of $2$ .

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Step 1.12.9.4.4.1

Cancel the common factor.

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - 2^{\frac{2}{2}}}{2} - 1$

Step 1.12.9.4.4.2

Rewrite the expression.

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - 2^{1}}{2} - 1$

Step 1.12.9.4.5

Evaluate the exponent.

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - 1 \cdot 2}{2} - 1$

Step 1.12.9.5

Multiply $- 1$ by $2$ .

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - 2}{2} - 1$

Step 1.13

Subtract $2$ from $6$ .

$\frac{4 + 2 \sqrt{3} - 2 \sqrt{3}}{2} - 1$

Step 1.14

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

$\frac{4 + 0}{2} - 1$

Step 1.15

Add $4$ and $0$ .

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

Step 1.16

Divide $4$ by $2$ .

$2 - 1$

Step 2

Subtract $1$ from $2$ .

$1$