Trigonometry Examples

$\cos (\frac{5 π}{8}) \cos (\frac{π}{8}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (\frac{5 π}{8})$ is $- \frac{\sqrt{2 - \sqrt{2}}}{2}$ .

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Step 1.1.1

Rewrite $\frac{5 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos (\frac{\frac{5 π}{4}}{2}) \cos (\frac{π}{8}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.1.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$(\pm \sqrt{\frac{1 + \cos (\frac{5 π}{4})}{2}}) \cos (\frac{π}{8}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.1.3

Change the $\pm$ to $-$ because cosine is negative in the second quadrant.

$- \sqrt{\frac{1 + \cos (\frac{5 π}{4})}{2}} \cos (\frac{π}{8}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.1.4

Simplify $- \sqrt{\frac{1 + \cos (\frac{5 π}{4})}{2}}$ .

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Step 1.1.4.1

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 third quadrant.

$- \sqrt{\frac{1 - \cos (\frac{π}{4})}{2}} \cos (\frac{π}{8}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.1.4.2

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

$- \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \cos (\frac{π}{8}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.1.4.3

Write $1$ as a fraction with a common denominator.

$- \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}} \cos (\frac{π}{8}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.1.4.4

Combine the numerators over the common denominator.

$- \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}} \cos (\frac{π}{8}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.1.4.5

Multiply the numerator by the reciprocal of the denominator.

$- \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}} \cos (\frac{π}{8}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.1.4.6

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

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Step 1.1.4.6.1

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

$- \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}} \cos (\frac{π}{8}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.1.4.6.2

Multiply $2$ by $2$ .

$- \sqrt{\frac{2 - \sqrt{2}}{4}} \cos (\frac{π}{8}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.1.4.7

Rewrite $\sqrt{\frac{2 - \sqrt{2}}{4}}$ as $\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

$- \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}} \cos (\frac{π}{8}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.1.4.8

Simplify the denominator.

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Step 1.1.4.8.1

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

$- \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}} \cos (\frac{π}{8}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.1.4.8.2

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

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \cos (\frac{π}{8}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.2

The exact value of $\cos (\frac{π}{8})$ is $\frac{\sqrt{2 + \sqrt{2}}}{2}$ .

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Step 1.2.1

Rewrite $\frac{π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \cos (\frac{\frac{π}{4}}{2}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.2.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$- \frac{\sqrt{2 - \sqrt{2}}}{2} (\pm \sqrt{\frac{1 + \cos (\frac{π}{4})}{2}}) + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.2.3

Change the $\pm$ to $+$ because cosine is positive in the first quadrant.

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \sqrt{\frac{1 + \cos (\frac{π}{4})}{2}} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.2.4

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

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.2.5

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

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Step 1.2.5.1

Write $1$ as a fraction with a common denominator.

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.2.5.2

Combine the numerators over the common denominator.

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.2.5.3

Multiply the numerator by the reciprocal of the denominator.

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.2.5.4

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

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Step 1.2.5.4.1

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

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.2.5.4.2

Multiply $2$ by $2$ .

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \sqrt{\frac{2 + \sqrt{2}}{4}} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.2.5.5

Rewrite $\sqrt{\frac{2 + \sqrt{2}}{4}}$ as $\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$ .

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \cdot \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.2.5.6

Simplify the denominator.

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Step 1.2.5.6.1

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

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \cdot \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.2.5.6.2

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

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \cdot \frac{\sqrt{2 + \sqrt{2}}}{2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3

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

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Step 1.3.1

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

$- \frac{\sqrt{2 + \sqrt{2}} \sqrt{2 - \sqrt{2}}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.2

Combine using the product rule for radicals.

$- \frac{\sqrt{(2 + \sqrt{2}) (2 - \sqrt{2})}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.3

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

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Step 1.3.3.1

Apply the distributive property.

$- \frac{\sqrt{2 (2 - \sqrt{2}) + \sqrt{2} (2 - \sqrt{2})}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.3.2

Apply the distributive property.

$- \frac{\sqrt{2 \cdot 2 + 2 (- \sqrt{2}) + \sqrt{2} (2 - \sqrt{2})}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.3.3

Apply the distributive property.

$- \frac{\sqrt{2 \cdot 2 + 2 (- \sqrt{2}) + \sqrt{2} \cdot 2 + \sqrt{2} (- \sqrt{2})}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4

Simplify and combine like terms.

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Step 1.3.4.1

Simplify each term.

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Step 1.3.4.1.1

Multiply $2$ by $2$ .

$- \frac{\sqrt{4 + 2 (- \sqrt{2}) + \sqrt{2} \cdot 2 + \sqrt{2} (- \sqrt{2})}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.1.2

Multiply $- 1$ by $2$ .

$- \frac{\sqrt{4 - 2 \sqrt{2} + \sqrt{2} \cdot 2 + \sqrt{2} (- \sqrt{2})}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.1.3

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

$- \frac{\sqrt{4 - 2 \sqrt{2} + 2 \cdot \sqrt{2} + \sqrt{2} (- \sqrt{2})}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.1.4

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

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Step 1.3.4.1.4.1

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

$- \frac{\sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - ({\sqrt{2}}^{1} \sqrt{2})}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.1.4.2

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

$- \frac{\sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.1.4.3

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

$- \frac{\sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - {\sqrt{2}}^{1 + 1}}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.1.4.4

Add $1$ and $1$ .

$- \frac{\sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - {\sqrt{2}}^{2}}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.1.5

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

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Step 1.3.4.1.5.1

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

$- \frac{\sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - {(2^{\frac{1}{2}})}^{2}}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.1.5.2

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

$- \frac{\sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - 2^{\frac{1}{2} \cdot 2}}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.1.5.3

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

$- \frac{\sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - 2^{\frac{2}{2}}}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.1.5.4

Cancel the common factor of $2$ .

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Step 1.3.4.1.5.4.1

Cancel the common factor.

$- \frac{\sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - 2^{\frac{2}{2}}}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.1.5.4.2

Rewrite the expression.

$- \frac{\sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - 2^{1}}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.1.5.5

Evaluate the exponent.

$- \frac{\sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - 1 \cdot 2}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.1.6

Multiply $- 1$ by $2$ .

$- \frac{\sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - 2}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.2

Subtract $2$ from $4$ .

$- \frac{\sqrt{2 - 2 \sqrt{2} + 2 \sqrt{2}}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.3

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

$- \frac{\sqrt{2 + 0}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.4.4

Add $2$ and $0$ .

$- \frac{\sqrt{2}}{2 \cdot 2} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.3.5

Multiply $2$ by $2$ .

$- \frac{\sqrt{2}}{4} + \sin (\frac{5 π}{8}) \sin (\frac{π}{8})$

Step 1.4

The exact value of $\sin (\frac{5 π}{8})$ is $\frac{\sqrt{2 + \sqrt{2}}}{2}$ .

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Step 1.4.1

Rewrite $\frac{5 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$- \frac{\sqrt{2}}{4} + \sin (\frac{\frac{5 π}{4}}{2}) \sin (\frac{π}{8})$

Step 1.4.2

Apply the sine half-angle identity.

$- \frac{\sqrt{2}}{4} + (\pm \sqrt{\frac{1 - \cos (\frac{5 π}{4})}{2}}) \sin (\frac{π}{8})$

Step 1.4.3

Change the $\pm$ to $+$ because sine is positive in the second quadrant.

$- \frac{\sqrt{2}}{4} + \sqrt{\frac{1 - \cos (\frac{5 π}{4})}{2}} \sin (\frac{π}{8})$

Step 1.4.4

Simplify $\sqrt{\frac{1 - \cos (\frac{5 π}{4})}{2}}$ .

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Step 1.4.4.1

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 third quadrant.

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

Step 1.4.4.2

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

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

Step 1.4.4.3

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

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Step 1.4.4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.4.4.3.2

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

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

Step 1.4.4.4

Write $1$ as a fraction with a common denominator.

$- \frac{\sqrt{2}}{4} + \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}} \sin (\frac{π}{8})$

Step 1.4.4.5

Combine the numerators over the common denominator.

$- \frac{\sqrt{2}}{4} + \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}} \sin (\frac{π}{8})$

Step 1.4.4.6

Multiply the numerator by the reciprocal of the denominator.

$- \frac{\sqrt{2}}{4} + \sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}} \sin (\frac{π}{8})$

Step 1.4.4.7

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

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Step 1.4.4.7.1

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

$- \frac{\sqrt{2}}{4} + \sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}} \sin (\frac{π}{8})$

Step 1.4.4.7.2

Multiply $2$ by $2$ .

$- \frac{\sqrt{2}}{4} + \sqrt{\frac{2 + \sqrt{2}}{4}} \sin (\frac{π}{8})$

Step 1.4.4.8

Rewrite $\sqrt{\frac{2 + \sqrt{2}}{4}}$ as $\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$ .

$- \frac{\sqrt{2}}{4} + \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}} \sin (\frac{π}{8})$

Step 1.4.4.9

Simplify the denominator.

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Step 1.4.4.9.1

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

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

Step 1.4.4.9.2

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

$- \frac{\sqrt{2}}{4} + \frac{\sqrt{2 + \sqrt{2}}}{2} \sin (\frac{π}{8})$

Step 1.5

The exact value of $\sin (\frac{π}{8})$ is $\frac{\sqrt{2 - \sqrt{2}}}{2}$ .

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Step 1.5.1

Rewrite $\frac{π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

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

Step 1.5.2

Apply the sine half-angle identity.

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

Step 1.5.3

Change the $\pm$ to $+$ because sine is positive in the first quadrant.

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

Step 1.5.4

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

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Step 1.5.4.1

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

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

Step 1.5.4.2

Write $1$ as a fraction with a common denominator.

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

Step 1.5.4.3

Combine the numerators over the common denominator.

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

Step 1.5.4.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5.4.5

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

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Step 1.5.4.5.1

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

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

Step 1.5.4.5.2

Multiply $2$ by $2$ .

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

Step 1.5.4.6

Rewrite $\sqrt{\frac{2 - \sqrt{2}}{4}}$ as $\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

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

Step 1.5.4.7

Simplify the denominator.

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Step 1.5.4.7.1

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

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

Step 1.5.4.7.2

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

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

Step 1.6

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

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Step 1.6.1

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

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

Step 1.6.2

Combine using the product rule for radicals.

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

Step 1.6.3

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

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Step 1.6.3.1

Apply the distributive property.

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

Step 1.6.3.2

Apply the distributive property.

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

Step 1.6.3.3

Apply the distributive property.

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

Step 1.6.4

Simplify and combine like terms.

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Step 1.6.4.1

Simplify each term.

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Step 1.6.4.1.1

Multiply $2$ by $2$ .

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

Step 1.6.4.1.2

Multiply $- 1$ by $2$ .

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

Step 1.6.4.1.3

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

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

Step 1.6.4.1.4

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

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Step 1.6.4.1.4.1

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

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

Step 1.6.4.1.4.2

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

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

Step 1.6.4.1.4.3

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

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

Step 1.6.4.1.4.4

Add $1$ and $1$ .

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

Step 1.6.4.1.5

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

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Step 1.6.4.1.5.1

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

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

Step 1.6.4.1.5.2

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

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

Step 1.6.4.1.5.3

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

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

Step 1.6.4.1.5.4

Cancel the common factor of $2$ .

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Step 1.6.4.1.5.4.1

Cancel the common factor.

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

Step 1.6.4.1.5.4.2

Rewrite the expression.

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

Step 1.6.4.1.5.5

Evaluate the exponent.

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

Step 1.6.4.1.6

Multiply $- 1$ by $2$ .

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

Step 1.6.4.2

Subtract $2$ from $4$ .

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

Step 1.6.4.3

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

$- \frac{\sqrt{2}}{4} + \frac{\sqrt{2 + 0}}{2 \cdot 2}$

Step 1.6.4.4

Add $2$ and $0$ .

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

Step 1.6.5

Multiply $2$ by $2$ .

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

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

$\frac{0}{4}$

Step 2.3

Divide $0$ by $4$ .

$0$