Precalculus Examples

$\cos (45) \cos (15) - \sin (45) \sin (15)$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{\sqrt{2}}{2} \cos (15) - \sin (45) \sin (15)$

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.

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

Step 1.2.2

Separate negation.

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

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

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

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

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

Step 1.2.8.1.1.2

Combine using the product rule for radicals.

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

Step 1.2.8.1.1.3

Multiply $2$ by $3$ .

$\frac{\sqrt{2}}{2} (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (45) \sin (15)$

Step 1.2.8.1.1.4

Multiply $2$ by $2$ .

$\frac{\sqrt{2}}{2} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (45) \sin (15)$

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 $\frac{1}{2}$ .

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

Step 1.2.8.1.2.2

Multiply $2$ by $2$ .

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

Step 1.2.8.2

Combine the numerators over the common denominator.

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Multiply $2$ by $4$ .

$\frac{\sqrt{2} (\sqrt{6} + \sqrt{2})}{8} - \sin (45) \sin (15)$

Step 1.4

Apply the distributive property.

$\frac{\sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{8} - \sin (45) \sin (15)$

Step 1.5

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 6} + \sqrt{2} \sqrt{2}}{8} - \sin (45) \sin (15)$

Step 1.6

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 6} + \sqrt{2 \cdot 2}}{8} - \sin (45) \sin (15)$

Step 1.7

Simplify each term.

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Step 1.7.1

Multiply $2$ by $6$ .

$\frac{\sqrt{12} + \sqrt{2 \cdot 2}}{8} - \sin (45) \sin (15)$

Step 1.7.2

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

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Step 1.7.2.1

Factor $4$ out of $12$ .

$\frac{\sqrt{4 (3)} + \sqrt{2 \cdot 2}}{8} - \sin (45) \sin (15)$

Step 1.7.2.2

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

$\frac{\sqrt{2^{2} \cdot 3} + \sqrt{2 \cdot 2}}{8} - \sin (45) \sin (15)$

Step 1.7.3

Pull terms out from under the radical.

$\frac{2 \sqrt{3} + \sqrt{2 \cdot 2}}{8} - \sin (45) \sin (15)$

Step 1.7.4

Multiply $2$ by $2$ .

$\frac{2 \sqrt{3} + \sqrt{4}}{8} - \sin (45) \sin (15)$

Step 1.7.5

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

$\frac{2 \sqrt{3} + \sqrt{2^{2}}}{8} - \sin (45) \sin (15)$

Step 1.7.6

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

$\frac{2 \sqrt{3} + 2}{8} - \sin (45) \sin (15)$

Step 1.8

Cancel the common factor of $2 \sqrt{3} + 2$ and $8$ .

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Step 1.8.1

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

$\frac{2 (\sqrt{3}) + 2}{8} - \sin (45) \sin (15)$

Step 1.8.2

Factor $2$ out of $2$ .

$\frac{2 (\sqrt{3}) + 2 (1)}{8} - \sin (45) \sin (15)$

Step 1.8.3

Factor $2$ out of $2 (\sqrt{3}) + 2 (1)$ .

$\frac{2 (\sqrt{3} + 1)}{8} - \sin (45) \sin (15)$

Step 1.8.4

Cancel the common factors.

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Step 1.8.4.1

Factor $2$ out of $8$ .

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

Step 1.8.4.2

Cancel the common factor.

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

Step 1.8.4.3

Rewrite the expression.

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

Step 1.9

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

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

Step 1.10

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

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Step 1.10.1

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

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

Step 1.10.2

Separate negation.

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

Step 1.10.3

Apply the difference of angles identity.

$\frac{\sqrt{3} + 1}{4} - \frac{\sqrt{2}}{2} (\sin (45) \cos (30) - \cos (45) \sin (30))$

Step 1.10.4

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

$\frac{\sqrt{3} + 1}{4} - \frac{\sqrt{2}}{2} (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30))$

Step 1.10.5

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

$\frac{\sqrt{3} + 1}{4} - \frac{\sqrt{2}}{2} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30))$

Step 1.10.6

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

$\frac{\sqrt{3} + 1}{4} - \frac{\sqrt{2}}{2} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30))$

Step 1.10.7

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

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

Step 1.10.8

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

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Step 1.10.8.1

Simplify each term.

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Step 1.10.8.1.1

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

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Step 1.10.8.1.1.1

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

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

Step 1.10.8.1.1.2

Combine using the product rule for radicals.

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

Step 1.10.8.1.1.3

Multiply $2$ by $3$ .

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

Step 1.10.8.1.1.4

Multiply $2$ by $2$ .

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

Step 1.10.8.1.2

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

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Step 1.10.8.1.2.1

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

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

Step 1.10.8.1.2.2

Multiply $2$ by $2$ .

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

Step 1.10.8.2

Combine the numerators over the common denominator.

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

Step 1.11

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

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Step 1.11.1

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

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

Step 1.11.2

Multiply $4$ by $2$ .

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

Step 1.12

Apply the distributive property.

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

Step 1.13

Combine using the product rule for radicals.

$\frac{\sqrt{3} + 1}{4} - \frac{\sqrt{6 \cdot 2} - \sqrt{2} \sqrt{2}}{8}$

Step 1.14

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

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Step 1.14.1

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

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

Step 1.14.2

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

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

Step 1.14.3

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

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

Step 1.14.4

Add $1$ and $1$ .

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

Step 1.15

Simplify each term.

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Step 1.15.1

Multiply $6$ by $2$ .

$\frac{\sqrt{3} + 1}{4} - \frac{\sqrt{12} - {\sqrt{2}}^{2}}{8}$

Step 1.15.2

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

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Step 1.15.2.1

Factor $4$ out of $12$ .

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

Step 1.15.2.2

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

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

Step 1.15.3

Pull terms out from under the radical.

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

Step 1.15.4

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

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Step 1.15.4.1

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

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

Step 1.15.4.2

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

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

Step 1.15.4.3

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

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

Step 1.15.4.4

Cancel the common factor of $2$ .

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Step 1.15.4.4.1

Cancel the common factor.

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

Step 1.15.4.4.2

Rewrite the expression.

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

Step 1.15.4.5

Evaluate the exponent.

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

Step 1.15.5

Multiply $- 1$ by $2$ .

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

Step 1.16

Cancel the common factor of $2 \sqrt{3} - 2$ and $8$ .

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Step 1.16.1

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

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

Step 1.16.2

Factor $2$ out of $- 2$ .

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

Step 1.16.3

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

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

Step 1.16.4

Cancel the common factors.

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Step 1.16.4.1

Factor $2$ out of $8$ .

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

Step 1.16.4.2

Cancel the common factor.

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

Step 1.16.4.3

Rewrite the expression.

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $- 1$ by $- 1$ .

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

Step 4

Simplify terms.

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Step 4.1

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

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

Step 4.2

Simplify by adding numbers.

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Step 4.2.1

Add $0$ and $1$ .

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

Step 4.2.2

Add $1$ and $1$ .

$\frac{2}{4}$

Step 4.3

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

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Step 4.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 4.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 4.3.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$