Trigonometry Examples

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

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

Step 1

Divide

2

$2$ by

2

$2$ .

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

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

Step 2

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 3

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

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

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

Step 4

Divide

2

$2$ by

2

$2$ .

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

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

Step 5

Any root of

1

$1$ is

1

$1$ .

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

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

Step 6

Add

1

$1$ and

1

$1$ .

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

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

Step 7

Rewrite

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

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

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

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

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

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

Step 8

Multiply

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

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

\frac{\sqrt{2}}{\sqrt{2}}

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

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

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

\frac{\sqrt{2}}{\sqrt{2}}

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

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

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

Step 9.2

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

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

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

Step 9.3

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

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

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 9.5

Add

1

$1$ and

1

$1$ .

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

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

Step 9.6

Rewrite

{\sqrt{2}}^{2}

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

2

$2$ .

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Step 9.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2}

$\sqrt{2}$ as

2^{\frac{1}{2}}

$2^{\frac{1}{2}}$ .

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

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 9.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 9.6.4

Cancel the common factor of

2

$2$ .

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Step 9.6.4.1

Cancel the common factor.

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

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

Step 9.6.4.2

Rewrite the expression.

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

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

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

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

Step 9.6.5

Evaluate the exponent.

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

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

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

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

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

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

Step 10

Combine using the product rule for radicals.

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

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

Step 11

Reorder factors in

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

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

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

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