Trigonometry Examples

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

Step 1

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

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

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

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

Step 8

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

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

Step 9

Expand the denominator using the FOIL method.

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

Step 10

Simplify.

$\sqrt{(2 - \sqrt{2}) \frac{2 - \sqrt{2}}{2}}$

Step 11

Apply the distributive property.

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

Step 12

Cancel the common factor of $2$ .

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Step 12.1

Cancel the common factor.

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

Step 12.2

Rewrite the expression.

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

Step 13

Combine $\frac{2 - \sqrt{2}}{2}$ and $\sqrt{2}$ .

$\sqrt{2 - \sqrt{2} - \frac{(2 - \sqrt{2}) \sqrt{2}}{2}}$

Step 14

Find the common denominator.

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Step 14.1

Write $2$ as a fraction with denominator $1$ .

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

Step 14.2

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

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

Step 14.3

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

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

Step 14.4

Write $- \sqrt{2}$ as a fraction with denominator $1$ .

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

Step 14.5

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

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

Step 14.6

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

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

Step 15

Combine the numerators over the common denominator.

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

Step 16

Simplify each term.

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Step 16.1

Multiply $2$ by $2$ .

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

Step 16.2

Multiply $2$ by $- 1$ .

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

Step 16.3

Apply the distributive property.

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

Step 16.4

Multiply $- 1$ by $2$ .

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

Step 16.5

Multiply $- - \sqrt{2}$ .

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Step 16.5.1

Multiply $- 1$ by $- 1$ .

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

Step 16.5.2

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

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

Step 16.6

Apply the distributive property.

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

Step 16.7

Combine using the product rule for radicals.

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

Step 16.8

Simplify each term.

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Step 16.8.1

Multiply $2$ by $2$ .

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

Step 16.8.2

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

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

Step 16.8.3

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

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

Step 17

Simplify terms.

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Step 17.1

Add $4$ and $2$ .

$\sqrt{\frac{6 - 2 \sqrt{2} - 2 \sqrt{2}}{2}}$

Step 17.2

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

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

Step 17.3

Cancel the common factor of $6 - 4 \sqrt{2}$ and $2$ .

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Step 17.3.1

Factor $2$ out of $6$ .

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

Step 17.3.2

Factor $2$ out of $- 4 \sqrt{2}$ .

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

Step 17.3.3

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

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

Step 17.3.4

Cancel the common factors.

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Step 17.3.4.1

Factor $2$ out of $2$ .

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

Step 17.3.4.2

Cancel the common factor.

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

Step 17.3.4.3

Rewrite the expression.

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

Step 17.3.4.4

Divide $3 - 2 \sqrt{2}$ by $1$ .

$\sqrt{3 - 2 \sqrt{2}}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$\sqrt{3 - 2 \sqrt{2}}$

Decimal Form:

$0.41421356 \dots$