Trigonometry Examples

$\frac{2 (\frac{1}{2}) - 2 (\frac{1}{4}) - 1}{1 - \sin^{2} (45 °)}$

Step 1

Apply pythagorean identity.

$\frac{2 (\frac{1}{2}) - 2 (\frac{1}{4}) - 1}{\cos^{2} (45 °)}$

Step 2

Simplify the numerator.

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Step 2.1

Cancel the common factor of $2$ .

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Step 2.1.1

Cancel the common factor.

$\frac{2 (\frac{1}{2}) - 2 (\frac{1}{4}) - 1}{\cos^{2} (45 °)}$

Step 2.1.2

Rewrite the expression.

$\frac{1 - 2 (\frac{1}{4}) - 1}{\cos^{2} (45 °)}$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Factor $2$ out of $- 2$ .

$\frac{1 + 2 (- 1) \frac{1}{4} - 1}{\cos^{2} (45 °)}$

Step 2.2.2

Factor $2$ out of $4$ .

$\frac{1 + 2 \cdot - 1 \frac{1}{2 \cdot 2} - 1}{\cos^{2} (45 °)}$

Step 2.2.3

Cancel the common factor.

$\frac{1 + 2 \cdot - 1 \frac{1}{2 \cdot 2} - 1}{\cos^{2} (45 °)}$

Step 2.2.4

Rewrite the expression.

$\frac{1 - 1 (\frac{1}{2}) - 1}{\cos^{2} (45 °)}$

Step 2.3

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

$\frac{1 - (\frac{1}{2}) - 1}{\cos^{2} (45 °)}$

Step 2.4

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

$\frac{\frac{2}{2} - \frac{1}{2} - 1}{\cos^{2} (45 °)}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{\frac{2 - 1}{2} - 1}{\cos^{2} (45 °)}$

Step 2.6

Subtract $1$ from $2$ .

$\frac{\frac{1}{2} - 1}{\cos^{2} (45 °)}$

Step 2.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{1}{2} - 1 \cdot \frac{2}{2}}{\cos^{2} (45 °)}$

Step 2.8

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

$\frac{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}{\cos^{2} (45 °)}$

Step 2.9

Combine the numerators over the common denominator.

$\frac{\frac{1 - 1 \cdot 2}{2}}{\cos^{2} (45 °)}$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $2$ .

$\frac{\frac{1 - 2}{2}}{\cos^{2} (45 °)}$

Step 2.10.2

Subtract $2$ from $1$ .

$\frac{\frac{- 1}{2}}{\cos^{2} (45 °)}$

Step 2.11

Move the negative in front of the fraction.

$\frac{- \frac{1}{2}}{\cos^{2} (45 °)}$

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Cancel the common factor of $2$ .

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Step 3.3.4.1

Cancel the common factor.

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

Step 3.3.4.2

Rewrite the expression.

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

Step 3.3.5

Evaluate the exponent.

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

Step 3.4

Raise $2$ to the power of $2$ .

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

Step 3.5

Cancel the common factor of $2$ and $4$ .

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Step 3.5.1

Factor $2$ out of $2$ .

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

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $2$ out of $4$ .

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

Step 3.5.2.2

Cancel the common factor.

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

Step 3.5.2.3

Rewrite the expression.

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

Step 4

Cancel the common factor of $\frac{1}{2}$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Divide $- 1$ by $1$ .

$- 1$