Trigonometry Examples

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

Step 1

Divide $2$ by $2$ .

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

Step 2

Multiply $- 1$ by $1$ .

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

Step 3

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 4

Divide $2$ by $2$ .

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

Step 5

Any root of $1$ is $1$ .

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

Step 6

Add $1$ and $1$ .

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

Step 7

Rewrite $\sqrt{\frac{1 - i}{2}}$ as $\frac{\sqrt{1 - i}}{\sqrt{2}}$ .

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

Step 8

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

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

Step 9

Combine and simplify the denominator.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

Add $1$ and $1$ .

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

Step 9.6

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

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Step 9.6.1

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

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

Step 9.6.2

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

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

Step 9.6.3

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

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

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

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

Step 9.6.4.2

Rewrite the expression.

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

Step 9.6.5

Evaluate the exponent.

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

Step 10

Combine using the product rule for radicals.

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

Step 11

Reorder factors in $\frac{\sqrt{(1 - i) \cdot 2}}{2}$ .

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