Evaluate tan(22.5 degrees )

$\tan (22.5 °)$

Step 1

Rewrite $22.5 °$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\tan (\frac{45}{2})$

Step 2

Apply the tangent half-angle identity.

$\pm \sqrt{\frac{1 - \cos (45)}{1 + \cos (45)}}$

Step 3

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

$\sqrt{\frac{1 - \cos (45)}{1 + \cos (45)}}$

Step 4

Simplify $\sqrt{\frac{1 - \cos (45)}{1 + \cos (45)}}$ .

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

$\sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{1 + \cos (45)}}$

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

$\sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{1 + \cos (45)}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{1 + \cos (45)}}$

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

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

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

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

Combine the numerators over the common denominator.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Apply the distributive property.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Simplify each term.

Apply the distributive property.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify each term.

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 1$ by $2$ .

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

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

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

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

Factor $2$ out of $- 2$ .

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

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

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $- 1$ .

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

Add $2$ and $1$ .

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

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.41421356 \dots$

Evaluate tan(22.5 degrees )

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