Precalculus Examples

$\sin (75) \cos (15) + \cos (105) \sin (15)$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\sin (75)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

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Step 1.1.1

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

$\sin (30 + 45) \cos (15) + \cos (105) \sin (15)$

Step 1.1.2

Apply the sum of angles identity.

$(\sin (30) \cos (45) + \cos (30) \sin (45)) \cos (15) + \cos (105) \sin (15)$

Step 1.1.3

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

$(\frac{1}{2} \cos (45) + \cos (30) \sin (45)) \cos (15) + \cos (105) \sin (15)$

Step 1.1.4

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)) \cos (15) + \cos (105) \sin (15)$

Step 1.1.5

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)) \cos (15) + \cos (105) \sin (15)$

Step 1.1.6

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \cos (15) + \cos (105) \sin (15)$

Step 1.1.7

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

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Step 1.1.7.1

Simplify each term.

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Step 1.1.7.1.1

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

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Step 1.1.7.1.1.1

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

$(\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \cos (15) + \cos (105) \sin (15)$

Step 1.1.7.1.1.2

Multiply $2$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \cos (15) + \cos (105) \sin (15)$

Step 1.1.7.1.2

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

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Step 1.1.7.1.2.1

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

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}) \cos (15) + \cos (105) \sin (15)$

Step 1.1.7.1.2.2

Combine using the product rule for radicals.

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}) \cos (15) + \cos (105) \sin (15)$

Step 1.1.7.1.2.3

Multiply $3$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}) \cos (15) + \cos (105) \sin (15)$

Step 1.1.7.1.2.4

Multiply $2$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}) \cos (15) + \cos (105) \sin (15)$

Step 1.1.7.2

Combine the numerators over the common denominator.

$\frac{\sqrt{2} + \sqrt{6}}{4} \cos (15) + \cos (105) \sin (15)$

Step 1.2

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

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Step 1.2.1

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

$\frac{\sqrt{2} + \sqrt{6}}{4} \cos (45 - 30) + \cos (105) \sin (15)$

Step 1.2.2

Separate negation.

$\frac{\sqrt{2} + \sqrt{6}}{4} \cos (45 - (30)) + \cos (105) \sin (15)$

Step 1.2.3

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

$\frac{\sqrt{2} + \sqrt{6}}{4} (\cos (45) \cos (30) + \sin (45) \sin (30)) + \cos (105) \sin (15)$

Step 1.2.4

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

$\frac{\sqrt{2} + \sqrt{6}}{4} (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30)) + \cos (105) \sin (15)$

Step 1.2.5

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

$\frac{\sqrt{2} + \sqrt{6}}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30)) + \cos (105) \sin (15)$

Step 1.2.6

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

$\frac{\sqrt{2} + \sqrt{6}}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30)) + \cos (105) \sin (15)$

Step 1.2.7

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

$\frac{\sqrt{2} + \sqrt{6}}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + \cos (105) \sin (15)$

Step 1.2.8

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

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Step 1.2.8.1

Simplify each term.

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Step 1.2.8.1.1

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

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Step 1.2.8.1.1.1

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

$\frac{\sqrt{2} + \sqrt{6}}{4} (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + \cos (105) \sin (15)$

Step 1.2.8.1.1.2

Combine using the product rule for radicals.

$\frac{\sqrt{2} + \sqrt{6}}{4} (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + \cos (105) \sin (15)$

Step 1.2.8.1.1.3

Multiply $2$ by $3$ .

$\frac{\sqrt{2} + \sqrt{6}}{4} (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + \cos (105) \sin (15)$

Step 1.2.8.1.1.4

Multiply $2$ by $2$ .

$\frac{\sqrt{2} + \sqrt{6}}{4} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + \cos (105) \sin (15)$

Step 1.2.8.1.2

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

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Step 1.2.8.1.2.1

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

$\frac{\sqrt{2} + \sqrt{6}}{4} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) + \cos (105) \sin (15)$

Step 1.2.8.1.2.2

Multiply $2$ by $2$ .

$\frac{\sqrt{2} + \sqrt{6}}{4} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) + \cos (105) \sin (15)$

Step 1.2.8.2

Combine the numerators over the common denominator.

$\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{\sqrt{6} + \sqrt{2}}{4} + \cos (105) \sin (15)$

Step 1.3

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 1.3.1

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

$\frac{(\sqrt{2} + \sqrt{6}) (\sqrt{6} + \sqrt{2})}{4 \cdot 4} + \cos (105) \sin (15)$

Step 1.3.2

Multiply $4$ by $4$ .

$\frac{(\sqrt{2} + \sqrt{6}) (\sqrt{6} + \sqrt{2})}{16} + \cos (105) \sin (15)$

Step 1.4

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

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Step 1.4.1

Apply the distributive property.

$\frac{\sqrt{2} (\sqrt{6} + \sqrt{2}) + \sqrt{6} (\sqrt{6} + \sqrt{2})}{16} + \cos (105) \sin (15)$

Step 1.4.2

Apply the distributive property.

$\frac{\sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2} + \sqrt{6} (\sqrt{6} + \sqrt{2})}{16} + \cos (105) \sin (15)$

Step 1.4.3

Apply the distributive property.

$\frac{\sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2} + \sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2}}{16} + \cos (105) \sin (15)$

Step 1.5

Simplify and combine like terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 6} + \sqrt{2} \sqrt{2} + \sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2}}{16} + \cos (105) \sin (15)$

Step 1.5.1.2

Multiply $2$ by $6$ .

$\frac{\sqrt{12} + \sqrt{2} \sqrt{2} + \sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2}}{16} + \cos (105) \sin (15)$

Step 1.5.1.3

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

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Step 1.5.1.3.1

Factor $4$ out of $12$ .

$\frac{\sqrt{4 (3)} + \sqrt{2} \sqrt{2} + \sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2}}{16} + \cos (105) \sin (15)$

Step 1.5.1.3.2

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

$\frac{\sqrt{2^{2} \cdot 3} + \sqrt{2} \sqrt{2} + \sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2}}{16} + \cos (105) \sin (15)$

Step 1.5.1.4

Pull terms out from under the radical.

$\frac{2 \sqrt{3} + \sqrt{2} \sqrt{2} + \sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2}}{16} + \cos (105) \sin (15)$

Step 1.5.1.5

Combine using the product rule for radicals.

$\frac{2 \sqrt{3} + \sqrt{2 \cdot 2} + \sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2}}{16} + \cos (105) \sin (15)$

Step 1.5.1.6

Multiply $2$ by $2$ .

$\frac{2 \sqrt{3} + \sqrt{4} + \sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2}}{16} + \cos (105) \sin (15)$

Step 1.5.1.7

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

$\frac{2 \sqrt{3} + \sqrt{2^{2}} + \sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2}}{16} + \cos (105) \sin (15)$

Step 1.5.1.8

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

$\frac{2 \sqrt{3} + 2 + \sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2}}{16} + \cos (105) \sin (15)$

Step 1.5.1.9

Combine using the product rule for radicals.

$\frac{2 \sqrt{3} + 2 + \sqrt{6 \cdot 6} + \sqrt{6} \sqrt{2}}{16} + \cos (105) \sin (15)$

Step 1.5.1.10

Multiply $6$ by $6$ .

$\frac{2 \sqrt{3} + 2 + \sqrt{36} + \sqrt{6} \sqrt{2}}{16} + \cos (105) \sin (15)$

Step 1.5.1.11

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

$\frac{2 \sqrt{3} + 2 + \sqrt{6^{2}} + \sqrt{6} \sqrt{2}}{16} + \cos (105) \sin (15)$

Step 1.5.1.12

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

$\frac{2 \sqrt{3} + 2 + 6 + \sqrt{6} \sqrt{2}}{16} + \cos (105) \sin (15)$

Step 1.5.1.13

Combine using the product rule for radicals.

$\frac{2 \sqrt{3} + 2 + 6 + \sqrt{6 \cdot 2}}{16} + \cos (105) \sin (15)$

Step 1.5.1.14

Multiply $6$ by $2$ .

$\frac{2 \sqrt{3} + 2 + 6 + \sqrt{12}}{16} + \cos (105) \sin (15)$

Step 1.5.1.15

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

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Step 1.5.1.15.1

Factor $4$ out of $12$ .

$\frac{2 \sqrt{3} + 2 + 6 + \sqrt{4 (3)}}{16} + \cos (105) \sin (15)$

Step 1.5.1.15.2

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

$\frac{2 \sqrt{3} + 2 + 6 + \sqrt{2^{2} \cdot 3}}{16} + \cos (105) \sin (15)$

Step 1.5.1.16

Pull terms out from under the radical.

$\frac{2 \sqrt{3} + 2 + 6 + 2 \sqrt{3}}{16} + \cos (105) \sin (15)$

Step 1.5.2

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

$\frac{2 + 6 + 4 \sqrt{3}}{16} + \cos (105) \sin (15)$

Step 1.5.3

Add $2$ and $6$ .

$\frac{8 + 4 \sqrt{3}}{16} + \cos (105) \sin (15)$

Step 1.6

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

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Step 1.6.1

Factor $4$ out of $8$ .

$\frac{4 \cdot 2 + 4 \sqrt{3}}{16} + \cos (105) \sin (15)$

Step 1.6.2

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

$\frac{4 \cdot 2 + 4 (\sqrt{3})}{16} + \cos (105) \sin (15)$

Step 1.6.3

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

$\frac{4 \cdot (2 + \sqrt{3})}{16} + \cos (105) \sin (15)$

Step 1.6.4

Cancel the common factors.

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Step 1.6.4.1

Factor $4$ out of $16$ .

$\frac{4 \cdot (2 + \sqrt{3})}{4 (4)} + \cos (105) \sin (15)$

Step 1.6.4.2

Cancel the common factor.

$\frac{4 \cdot (2 + \sqrt{3})}{4 \cdot 4} + \cos (105) \sin (15)$

Step 1.6.4.3

Rewrite the expression.

$\frac{2 + \sqrt{3}}{4} + \cos (105) \sin (15)$

Step 1.7

The exact value of $\cos (105)$ is $- \frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.7.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{2 + \sqrt{3}}{4} - \cos (75) \sin (15)$

Step 1.7.2

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

$\frac{2 + \sqrt{3}}{4} - \cos (30 + 45) \sin (15)$

Step 1.7.3

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

$\frac{2 + \sqrt{3}}{4} - (\cos (30) \cos (45) - \sin (30) \sin (45)) \sin (15)$

Step 1.7.4

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

$\frac{2 + \sqrt{3}}{4} - (\frac{\sqrt{3}}{2} \cos (45) - \sin (30) \sin (45)) \sin (15)$

Step 1.7.5

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

$\frac{2 + \sqrt{3}}{4} - (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (30) \sin (45)) \sin (15)$

Step 1.7.6

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

$\frac{2 + \sqrt{3}}{4} - (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (45)) \sin (15)$

Step 1.7.7

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

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

Step 1.7.8

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

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Step 1.7.8.1

Simplify each term.

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Step 1.7.8.1.1

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

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Step 1.7.8.1.1.1

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

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

Step 1.7.8.1.1.2

Combine using the product rule for radicals.

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

Step 1.7.8.1.1.3

Multiply $3$ by $2$ .

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

Step 1.7.8.1.1.4

Multiply $2$ by $2$ .

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

Step 1.7.8.1.2

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

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Step 1.7.8.1.2.1

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

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

Step 1.7.8.1.2.2

Multiply $2$ by $2$ .

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

Step 1.7.8.2

Combine the numerators over the common denominator.

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

Step 1.8

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

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Step 1.8.1

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

$\frac{2 + \sqrt{3}}{4} - \frac{\sqrt{6} - \sqrt{2}}{4} \sin (45 - 30)$

Step 1.8.2

Separate negation.

$\frac{2 + \sqrt{3}}{4} - \frac{\sqrt{6} - \sqrt{2}}{4} \sin (45 - (30))$

Step 1.8.3

Apply the difference of angles identity.

$\frac{2 + \sqrt{3}}{4} - \frac{\sqrt{6} - \sqrt{2}}{4} (\sin (45) \cos (30) - \cos (45) \sin (30))$

Step 1.8.4

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

$\frac{2 + \sqrt{3}}{4} - \frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30))$

Step 1.8.5

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

$\frac{2 + \sqrt{3}}{4} - \frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30))$

Step 1.8.6

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

$\frac{2 + \sqrt{3}}{4} - \frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30))$

Step 1.8.7

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

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

Step 1.8.8

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

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Step 1.8.8.1

Simplify each term.

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Step 1.8.8.1.1

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

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Step 1.8.8.1.1.1

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

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

Step 1.8.8.1.1.2

Combine using the product rule for radicals.

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

Step 1.8.8.1.1.3

Multiply $2$ by $3$ .

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

Step 1.8.8.1.1.4

Multiply $2$ by $2$ .

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

Step 1.8.8.1.2

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

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Step 1.8.8.1.2.1

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

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

Step 1.8.8.1.2.2

Multiply $2$ by $2$ .

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

Step 1.8.8.2

Combine the numerators over the common denominator.

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

Step 1.9

Multiply $- \frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.9.1

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

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

Step 1.9.2

Raise $\sqrt{6} - \sqrt{2}$ to the power of $1$ .

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

Step 1.9.3

Raise $\sqrt{6} - \sqrt{2}$ to the power of $1$ .

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

Step 1.9.4

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

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

Step 1.9.5

Add $1$ and $1$ .

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

Step 1.9.6

Multiply $4$ by $4$ .

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

Step 1.10

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

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

Step 1.11

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

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Step 1.11.1

Apply the distributive property.

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

Step 1.11.2

Apply the distributive property.

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

Step 1.11.3

Apply the distributive property.

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

Step 1.12

Simplify and combine like terms.

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Step 1.12.1

Simplify each term.

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Step 1.12.1.1

Combine using the product rule for radicals.

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

Step 1.12.1.2

Multiply $6$ by $6$ .

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

Step 1.12.1.3

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

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

Step 1.12.1.4

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

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

Step 1.12.1.5

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

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Step 1.12.1.5.1

Combine using the product rule for radicals.

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

Step 1.12.1.5.2

Multiply $6$ by $2$ .

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

Step 1.12.1.6

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

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Step 1.12.1.6.1

Factor $4$ out of $12$ .

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

Step 1.12.1.6.2

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

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

Step 1.12.1.7

Pull terms out from under the radical.

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

Step 1.12.1.8

Multiply $2$ by $- 1$ .

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

Step 1.12.1.9

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

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Step 1.12.1.9.1

Combine using the product rule for radicals.

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

Step 1.12.1.9.2

Multiply $6$ by $2$ .

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

Step 1.12.1.10

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

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Step 1.12.1.10.1

Factor $4$ out of $12$ .

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

Step 1.12.1.10.2

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

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

Step 1.12.1.11

Pull terms out from under the radical.

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

Step 1.12.1.12

Multiply $2$ by $- 1$ .

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

Step 1.12.1.13

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

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Step 1.12.1.13.1

Multiply $- 1$ by $- 1$ .

$\frac{2 + \sqrt{3}}{4} - \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 1 \sqrt{2} \sqrt{2}}{16}$

Step 1.12.1.13.2

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

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

Step 1.12.1.13.3

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

$\frac{2 + \sqrt{3}}{4} - \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1} \sqrt{2}}{16}$

Step 1.12.1.13.4

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

$\frac{2 + \sqrt{3}}{4} - \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1} {\sqrt{2}}^{1}}{16}$

Step 1.12.1.13.5

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

$\frac{2 + \sqrt{3}}{4} - \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1 + 1}}{16}$

Step 1.12.1.13.6

Add $1$ and $1$ .

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

Step 1.12.1.14

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

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Step 1.12.1.14.1

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

$\frac{2 + \sqrt{3}}{4} - \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {(2^{\frac{1}{2}})}^{2}}{16}$

Step 1.12.1.14.2

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

$\frac{2 + \sqrt{3}}{4} - \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{1}{2} \cdot 2}}{16}$

Step 1.12.1.14.3

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

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

Step 1.12.1.14.4

Cancel the common factor of $2$ .

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Step 1.12.1.14.4.1

Cancel the common factor.

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

Step 1.12.1.14.4.2

Rewrite the expression.

$\frac{2 + \sqrt{3}}{4} - \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{1}}{16}$

Step 1.12.1.14.5

Evaluate the exponent.

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

Step 1.12.2

Add $6$ and $2$ .

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

Step 1.12.3

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

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

Step 1.13

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

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Step 1.13.1

Factor $4$ out of $8$ .

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

Step 1.13.2

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

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

Step 1.13.3

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

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

Step 1.13.4

Cancel the common factors.

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Step 1.13.4.1

Factor $4$ out of $16$ .

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

Step 1.13.4.2

Cancel the common factor.

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

Step 1.13.4.3

Rewrite the expression.

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $- 1$ by $2$ .

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

Step 3.3

Multiply $- - \sqrt{3}$ .

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Step 3.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.2

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

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

Step 4

Simplify terms.

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Step 4.1

Subtract $2$ from $2$ .

$\frac{0 + \sqrt{3} + \sqrt{3}}{4}$

Step 4.2

Add $0$ and $\sqrt{3}$ .

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

Step 4.3

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

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

Step 4.4

Cancel the common factor of $2$ and $4$ .

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Step 4.4.1

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

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

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $4$ .

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

Step 4.4.2.2

Cancel the common factor.

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

Step 4.4.2.3

Rewrite the expression.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.86602540 \dots$