Trigonometry Examples

$\sqrt{3} - \cos^{2} (15) - \sqrt{3} \sin^{2} (15)$

Step 1

Move $- \sqrt{3} \sin^{2} (15)$ .

$\sqrt{3} - \sqrt{3} \sin^{2} (15) - \cos^{2} (15)$

Step 2

Multiply by $1$ .

$\sqrt{3} \cdot 1 - \sqrt{3} \sin^{2} (15) - \cos^{2} (15)$

Step 3

Simplify with factoring out.

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Step 3.1

Factor $\sqrt{3}$ out of $- \sqrt{3} \sin^{2} (15)$ .

$\sqrt{3} \cdot 1 + \sqrt{3} (- 1 \sin^{2} (15)) - \cos^{2} (15)$

Step 3.2

Factor $\sqrt{3}$ out of $\sqrt{3} \cdot 1 + \sqrt{3} (- 1 \sin^{2} (15))$ .

$\sqrt{3} \cdot (1 - 1 \sin^{2} (15)) - \cos^{2} (15)$

Step 3.3

Rewrite $- 1 \sin^{2} (15)$ as $- \sin^{2} (15)$ .

$\sqrt{3} \cdot (1 - \sin^{2} (15)) - \cos^{2} (15)$

Step 4

Apply pythagorean identity.

$\sqrt{3} \cdot \cos^{2} (15) - \cos^{2} (15)$

Step 5

Simplify terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

The exact value of $\cos (15)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 5.1.1.1

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

$\sqrt{3} \cdot \cos^{2} (45 - 30) - \cos^{2} (15)$

Step 5.1.1.2

Separate negation.

$\sqrt{3} \cdot \cos^{2} (45 - (30)) - \cos^{2} (15)$

Step 5.1.1.3

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

$\sqrt{3} \cdot {(\cos (45) \cos (30) + \sin (45) \sin (30))}^{2} - \cos^{2} (15)$

Step 5.1.1.4

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

$\sqrt{3} \cdot {(\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30))}^{2} - \cos^{2} (15)$

Step 5.1.1.5

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

$\sqrt{3} \cdot {(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30))}^{2} - \cos^{2} (15)$

Step 5.1.1.6

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

$\sqrt{3} \cdot {(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30))}^{2} - \cos^{2} (15)$

Step 5.1.1.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

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

Step 5.1.1.8

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

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Step 5.1.1.8.1

Simplify each term.

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Step 5.1.1.8.1.1

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

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Step 5.1.1.8.1.1.1

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

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

Step 5.1.1.8.1.1.2

Combine using the product rule for radicals.

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

Step 5.1.1.8.1.1.3

Multiply $2$ by $3$ .

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

Step 5.1.1.8.1.1.4

Multiply $2$ by $2$ .

$\sqrt{3} \cdot {(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})}^{2} - \cos^{2} (15)$

Step 5.1.1.8.1.2

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

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Step 5.1.1.8.1.2.1

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

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

Step 5.1.1.8.1.2.2

Multiply $2$ by $2$ .

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

Step 5.1.1.8.2

Combine the numerators over the common denominator.

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

Step 5.1.2

Apply the product rule to $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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

Step 5.1.3

Raise $4$ to the power of $2$ .

$\sqrt{3} \cdot \frac{{(\sqrt{6} + \sqrt{2})}^{2}}{16} - \cos^{2} (15)$

Step 5.1.4

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

$\sqrt{3} \cdot \frac{(\sqrt{6} + \sqrt{2}) (\sqrt{6} + \sqrt{2})}{16} - \cos^{2} (15)$

Step 5.1.5

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

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Step 5.1.5.1

Apply the distributive property.

$\sqrt{3} \cdot \frac{\sqrt{6} (\sqrt{6} + \sqrt{2}) + \sqrt{2} (\sqrt{6} + \sqrt{2})}{16} - \cos^{2} (15)$

Step 5.1.5.2

Apply the distributive property.

$\sqrt{3} \cdot \frac{\sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2} + \sqrt{2} (\sqrt{6} + \sqrt{2})}{16} - \cos^{2} (15)$

Step 5.1.5.3

Apply the distributive property.

$\sqrt{3} \cdot \frac{\sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16} - \cos^{2} (15)$

Step 5.1.6

Simplify and combine like terms.

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Step 5.1.6.1

Simplify each term.

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Step 5.1.6.1.1

Combine using the product rule for radicals.

$\sqrt{3} \cdot \frac{\sqrt{6 \cdot 6} + \sqrt{6} \sqrt{2} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16} - \cos^{2} (15)$

Step 5.1.6.1.2

Multiply $6$ by $6$ .

$\sqrt{3} \cdot \frac{\sqrt{36} + \sqrt{6} \sqrt{2} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16} - \cos^{2} (15)$

Step 5.1.6.1.3

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

$\sqrt{3} \cdot \frac{\sqrt{6^{2}} + \sqrt{6} \sqrt{2} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16} - \cos^{2} (15)$

Step 5.1.6.1.4

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

$\sqrt{3} \cdot \frac{6 + \sqrt{6} \sqrt{2} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16} - \cos^{2} (15)$

Step 5.1.6.1.5

Combine using the product rule for radicals.

$\sqrt{3} \cdot \frac{6 + \sqrt{6 \cdot 2} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16} - \cos^{2} (15)$

Step 5.1.6.1.6

Multiply $6$ by $2$ .

$\sqrt{3} \cdot \frac{6 + \sqrt{12} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16} - \cos^{2} (15)$

Step 5.1.6.1.7

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

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Step 5.1.6.1.7.1

Factor $4$ out of $12$ .

$\sqrt{3} \cdot \frac{6 + \sqrt{4 (3)} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16} - \cos^{2} (15)$

Step 5.1.6.1.7.2

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

$\sqrt{3} \cdot \frac{6 + \sqrt{2^{2} \cdot 3} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16} - \cos^{2} (15)$

Step 5.1.6.1.8

Pull terms out from under the radical.

$\sqrt{3} \cdot \frac{6 + 2 \sqrt{3} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16} - \cos^{2} (15)$

Step 5.1.6.1.9

Combine using the product rule for radicals.

$\sqrt{3} \cdot \frac{6 + 2 \sqrt{3} + \sqrt{2 \cdot 6} + \sqrt{2} \sqrt{2}}{16} - \cos^{2} (15)$

Step 5.1.6.1.10

Multiply $2$ by $6$ .

$\sqrt{3} \cdot \frac{6 + 2 \sqrt{3} + \sqrt{12} + \sqrt{2} \sqrt{2}}{16} - \cos^{2} (15)$

Step 5.1.6.1.11

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

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Step 5.1.6.1.11.1

Factor $4$ out of $12$ .

$\sqrt{3} \cdot \frac{6 + 2 \sqrt{3} + \sqrt{4 (3)} + \sqrt{2} \sqrt{2}}{16} - \cos^{2} (15)$

Step 5.1.6.1.11.2

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

$\sqrt{3} \cdot \frac{6 + 2 \sqrt{3} + \sqrt{2^{2} \cdot 3} + \sqrt{2} \sqrt{2}}{16} - \cos^{2} (15)$

Step 5.1.6.1.12

Pull terms out from under the radical.

$\sqrt{3} \cdot \frac{6 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{2} \sqrt{2}}{16} - \cos^{2} (15)$

Step 5.1.6.1.13

Combine using the product rule for radicals.

$\sqrt{3} \cdot \frac{6 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{2 \cdot 2}}{16} - \cos^{2} (15)$

Step 5.1.6.1.14

Multiply $2$ by $2$ .

$\sqrt{3} \cdot \frac{6 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{4}}{16} - \cos^{2} (15)$

Step 5.1.6.1.15

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

$\sqrt{3} \cdot \frac{6 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{2^{2}}}{16} - \cos^{2} (15)$

Step 5.1.6.1.16

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

$\sqrt{3} \cdot \frac{6 + 2 \sqrt{3} + 2 \sqrt{3} + 2}{16} - \cos^{2} (15)$

Step 5.1.6.2

Add $6$ and $2$ .

$\sqrt{3} \cdot \frac{8 + 2 \sqrt{3} + 2 \sqrt{3}}{16} - \cos^{2} (15)$

Step 5.1.6.3

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

$\sqrt{3} \cdot \frac{8 + 4 \sqrt{3}}{16} - \cos^{2} (15)$

Step 5.1.7

Cancel the common factor of $8 + 4 \sqrt{3}$ and $16$ .

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Step 5.1.7.1

Factor $4$ out of $8$ .

$\sqrt{3} \cdot \frac{4 \cdot 2 + 4 \sqrt{3}}{16} - \cos^{2} (15)$

Step 5.1.7.2

Factor $4$ out of $4 \sqrt{3}$ .

$\sqrt{3} \cdot \frac{4 \cdot 2 + 4 (\sqrt{3})}{16} - \cos^{2} (15)$

Step 5.1.7.3

Factor $4$ out of $4 \cdot 2 + 4 (\sqrt{3})$ .

$\sqrt{3} \cdot \frac{4 \cdot (2 + \sqrt{3})}{16} - \cos^{2} (15)$

Step 5.1.7.4

Cancel the common factors.

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Step 5.1.7.4.1

Factor $4$ out of $16$ .

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

Step 5.1.7.4.2

Cancel the common factor.

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

Step 5.1.7.4.3

Rewrite the expression.

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

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

Step 5.1.8

Apply the distributive property.

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

Step 5.1.9

Move $2$ to the left of $\sqrt{3}$ .

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

Step 5.1.10

Combine using the product rule for radicals.

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

Step 5.1.11

Simplify each term.

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Step 5.1.11.1

Multiply $3$ by $3$ .

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

Step 5.1.11.2

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

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

Step 5.1.11.3

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

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

Step 5.1.12

The exact value of $\cos (15)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 5.1.12.1

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

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

Step 5.1.12.2

Separate negation.

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

Step 5.1.12.3

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

$\frac{2 \sqrt{3} + 3}{4} - {(\cos (45) \cos (30) + \sin (45) \sin (30))}^{2}$

Step 5.1.12.4

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

$\frac{2 \sqrt{3} + 3}{4} - {(\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30))}^{2}$

Step 5.1.12.5

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

$\frac{2 \sqrt{3} + 3}{4} - {(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30))}^{2}$

Step 5.1.12.6

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

$\frac{2 \sqrt{3} + 3}{4} - {(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30))}^{2}$

Step 5.1.12.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

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

Step 5.1.12.8

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

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Step 5.1.12.8.1

Simplify each term.

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Step 5.1.12.8.1.1

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

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Step 5.1.12.8.1.1.1

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

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

Step 5.1.12.8.1.1.2

Combine using the product rule for radicals.

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

Step 5.1.12.8.1.1.3

Multiply $2$ by $3$ .

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

Step 5.1.12.8.1.1.4

Multiply $2$ by $2$ .

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

Step 5.1.12.8.1.2

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

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Step 5.1.12.8.1.2.1

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

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

Step 5.1.12.8.1.2.2

Multiply $2$ by $2$ .

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

Step 5.1.12.8.2

Combine the numerators over the common denominator.

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

Step 5.1.13

Apply the product rule to $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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

Step 5.1.14

Raise $4$ to the power of $2$ .

$\frac{2 \sqrt{3} + 3}{4} - \frac{{(\sqrt{6} + \sqrt{2})}^{2}}{16}$

Step 5.1.15

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

$\frac{2 \sqrt{3} + 3}{4} - \frac{(\sqrt{6} + \sqrt{2}) (\sqrt{6} + \sqrt{2})}{16}$

Step 5.1.16

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

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Step 5.1.16.1

Apply the distributive property.

$\frac{2 \sqrt{3} + 3}{4} - \frac{\sqrt{6} (\sqrt{6} + \sqrt{2}) + \sqrt{2} (\sqrt{6} + \sqrt{2})}{16}$

Step 5.1.16.2

Apply the distributive property.

$\frac{2 \sqrt{3} + 3}{4} - \frac{\sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2} + \sqrt{2} (\sqrt{6} + \sqrt{2})}{16}$

Step 5.1.16.3

Apply the distributive property.

$\frac{2 \sqrt{3} + 3}{4} - \frac{\sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16}$

Step 5.1.17

Simplify and combine like terms.

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Step 5.1.17.1

Simplify each term.

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Step 5.1.17.1.1

Combine using the product rule for radicals.

$\frac{2 \sqrt{3} + 3}{4} - \frac{\sqrt{6 \cdot 6} + \sqrt{6} \sqrt{2} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16}$

Step 5.1.17.1.2

Multiply $6$ by $6$ .

$\frac{2 \sqrt{3} + 3}{4} - \frac{\sqrt{36} + \sqrt{6} \sqrt{2} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16}$

Step 5.1.17.1.3

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

$\frac{2 \sqrt{3} + 3}{4} - \frac{\sqrt{6^{2}} + \sqrt{6} \sqrt{2} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16}$

Step 5.1.17.1.4

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

$\frac{2 \sqrt{3} + 3}{4} - \frac{6 + \sqrt{6} \sqrt{2} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16}$

Step 5.1.17.1.5

Combine using the product rule for radicals.

$\frac{2 \sqrt{3} + 3}{4} - \frac{6 + \sqrt{6 \cdot 2} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16}$

Step 5.1.17.1.6

Multiply $6$ by $2$ .

$\frac{2 \sqrt{3} + 3}{4} - \frac{6 + \sqrt{12} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16}$

Step 5.1.17.1.7

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

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Step 5.1.17.1.7.1

Factor $4$ out of $12$ .

$\frac{2 \sqrt{3} + 3}{4} - \frac{6 + \sqrt{4 (3)} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16}$

Step 5.1.17.1.7.2

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

$\frac{2 \sqrt{3} + 3}{4} - \frac{6 + \sqrt{2^{2} \cdot 3} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16}$

Step 5.1.17.1.8

Pull terms out from under the radical.

$\frac{2 \sqrt{3} + 3}{4} - \frac{6 + 2 \sqrt{3} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{16}$

Step 5.1.17.1.9

Combine using the product rule for radicals.

$\frac{2 \sqrt{3} + 3}{4} - \frac{6 + 2 \sqrt{3} + \sqrt{2 \cdot 6} + \sqrt{2} \sqrt{2}}{16}$

Step 5.1.17.1.10

Multiply $2$ by $6$ .

$\frac{2 \sqrt{3} + 3}{4} - \frac{6 + 2 \sqrt{3} + \sqrt{12} + \sqrt{2} \sqrt{2}}{16}$

Step 5.1.17.1.11

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

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Step 5.1.17.1.11.1

Factor $4$ out of $12$ .

$\frac{2 \sqrt{3} + 3}{4} - \frac{6 + 2 \sqrt{3} + \sqrt{4 (3)} + \sqrt{2} \sqrt{2}}{16}$

Step 5.1.17.1.11.2

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

$\frac{2 \sqrt{3} + 3}{4} - \frac{6 + 2 \sqrt{3} + \sqrt{2^{2} \cdot 3} + \sqrt{2} \sqrt{2}}{16}$

Step 5.1.17.1.12

Pull terms out from under the radical.

$\frac{2 \sqrt{3} + 3}{4} - \frac{6 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{2} \sqrt{2}}{16}$

Step 5.1.17.1.13

Combine using the product rule for radicals.

$\frac{2 \sqrt{3} + 3}{4} - \frac{6 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{2 \cdot 2}}{16}$

Step 5.1.17.1.14

Multiply $2$ by $2$ .

$\frac{2 \sqrt{3} + 3}{4} - \frac{6 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{4}}{16}$

Step 5.1.17.1.15

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

$\frac{2 \sqrt{3} + 3}{4} - \frac{6 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{2^{2}}}{16}$

Step 5.1.17.1.16

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

$\frac{2 \sqrt{3} + 3}{4} - \frac{6 + 2 \sqrt{3} + 2 \sqrt{3} + 2}{16}$

Step 5.1.17.2

Add $6$ and $2$ .

$\frac{2 \sqrt{3} + 3}{4} - \frac{8 + 2 \sqrt{3} + 2 \sqrt{3}}{16}$

Step 5.1.17.3

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

$\frac{2 \sqrt{3} + 3}{4} - \frac{8 + 4 \sqrt{3}}{16}$

Step 5.1.18

Cancel the common factor of $8 + 4 \sqrt{3}$ and $16$ .

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Step 5.1.18.1

Factor $4$ out of $8$ .

$\frac{2 \sqrt{3} + 3}{4} - \frac{4 \cdot 2 + 4 \sqrt{3}}{16}$

Step 5.1.18.2

Factor $4$ out of $4 \sqrt{3}$ .

$\frac{2 \sqrt{3} + 3}{4} - \frac{4 \cdot 2 + 4 (\sqrt{3})}{16}$

Step 5.1.18.3

Factor $4$ out of $4 \cdot 2 + 4 (\sqrt{3})$ .

$\frac{2 \sqrt{3} + 3}{4} - \frac{4 \cdot (2 + \sqrt{3})}{16}$

Step 5.1.18.4

Cancel the common factors.

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Step 5.1.18.4.1

Factor $4$ out of $16$ .

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

Step 5.1.18.4.2

Cancel the common factor.

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

Step 5.1.18.4.3

Rewrite the expression.

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 5.3

Simplify each term.

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Step 5.3.1

Apply the distributive property.

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

Step 5.3.2

Multiply $- 1$ by $2$ .

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

Step 5.4

Simplify by adding terms.

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Step 5.4.1

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

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

Step 5.4.2

Subtract $2$ from $3$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.68301270 \dots$