Trigonometry Examples

$\cos^{2} (112.5) - \sin^{2} (112.5)$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (112.5)$ is $- \frac{\sqrt{2 - \sqrt{2}}}{2}$ .

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Step 1.1.1

Rewrite $112.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos^{2} (\frac{225}{2}) - \sin^{2} (112.5)$

Step 1.1.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

${(\pm \sqrt{\frac{1 + \cos (225)}{2}})}^{2} - \sin^{2} (112.5)$

Step 1.1.3

Change the $\pm$ to $-$ because cosine is negative in the second quadrant.

${(- \sqrt{\frac{1 + \cos (225)}{2}})}^{2} - \sin^{2} (112.5)$

Step 1.1.4

Simplify $- \sqrt{\frac{1 + \cos (225)}{2}}$ .

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Step 1.1.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

${(- \sqrt{\frac{1 - \cos (45)}{2}})}^{2} - \sin^{2} (112.5)$

Step 1.1.4.2

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

${(- \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}})}^{2} - \sin^{2} (112.5)$

Step 1.1.4.3

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

${(- \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}})}^{2} - \sin^{2} (112.5)$

Step 1.1.4.4

Combine the numerators over the common denominator.

${(- \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}})}^{2} - \sin^{2} (112.5)$

Step 1.1.4.5

Multiply the numerator by the reciprocal of the denominator.

${(- \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}})}^{2} - \sin^{2} (112.5)$

Step 1.1.4.6

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

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Step 1.1.4.6.1

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

${(- \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}})}^{2} - \sin^{2} (112.5)$

Step 1.1.4.6.2

Multiply $2$ by $2$ .

${(- \sqrt{\frac{2 - \sqrt{2}}{4}})}^{2} - \sin^{2} (112.5)$

Step 1.1.4.7

Rewrite $\sqrt{\frac{2 - \sqrt{2}}{4}}$ as $\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

${(- \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}})}^{2} - \sin^{2} (112.5)$

Step 1.1.4.8

Simplify the denominator.

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Step 1.1.4.8.1

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

${(- \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}})}^{2} - \sin^{2} (112.5)$

Step 1.1.4.8.2

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

${(- \frac{\sqrt{2 - \sqrt{2}}}{2})}^{2} - \sin^{2} (112.5)$

Step 1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.2.1

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

${(- 1)}^{2} {(\frac{\sqrt{2 - \sqrt{2}}}{2})}^{2} - \sin^{2} (112.5)$

Step 1.2.2

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

${(- 1)}^{2} \frac{{\sqrt{2 - \sqrt{2}}}^{2}}{2^{2}} - \sin^{2} (112.5)$

Step 1.3

Raise $- 1$ to the power of $2$ .

$1 \frac{{\sqrt{2 - \sqrt{2}}}^{2}}{2^{2}} - \sin^{2} (112.5)$

Step 1.4

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

$\frac{{\sqrt{2 - \sqrt{2}}}^{2}}{2^{2}} - \sin^{2} (112.5)$

Step 1.5

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

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Step 1.5.1

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

$\frac{{({(2 - \sqrt{2})}^{\frac{1}{2}})}^{2}}{2^{2}} - \sin^{2} (112.5)$

Step 1.5.2

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

$\frac{{(2 - \sqrt{2})}^{\frac{1}{2} \cdot 2}}{2^{2}} - \sin^{2} (112.5)$

Step 1.5.3

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

$\frac{{(2 - \sqrt{2})}^{\frac{2}{2}}}{2^{2}} - \sin^{2} (112.5)$

Step 1.5.4

Cancel the common factor of $2$ .

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Step 1.5.4.1

Cancel the common factor.

$\frac{{(2 - \sqrt{2})}^{\frac{2}{2}}}{2^{2}} - \sin^{2} (112.5)$

Step 1.5.4.2

Rewrite the expression.

$\frac{{(2 - \sqrt{2})}^{1}}{2^{2}} - \sin^{2} (112.5)$

Step 1.5.5

Simplify.

$\frac{2 - \sqrt{2}}{2^{2}} - \sin^{2} (112.5)$

Step 1.6

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

$\frac{2 - \sqrt{2}}{4} - \sin^{2} (112.5)$

Step 1.7

The exact value of $\sin (112.5)$ is $\frac{\sqrt{2 + \sqrt{2}}}{2}$ .

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Step 1.7.1

Rewrite $112.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\frac{2 - \sqrt{2}}{4} - \sin^{2} (\frac{225}{2})$

Step 1.7.2

Apply the sine half-angle identity.

$\frac{2 - \sqrt{2}}{4} - {(\pm \sqrt{\frac{1 - \cos (225)}{2}})}^{2}$

Step 1.7.3

Change the $\pm$ to $+$ because sine is positive in the second quadrant.

$\frac{2 - \sqrt{2}}{4} - {\sqrt{\frac{1 - \cos (225)}{2}}}^{2}$

Step 1.7.4

Simplify $\sqrt{\frac{1 - \cos (225)}{2}}$ .

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Step 1.7.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

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

Step 1.7.4.2

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

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

Step 1.7.4.3

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

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Step 1.7.4.3.1

Multiply $- 1$ by $- 1$ .

$\frac{2 - \sqrt{2}}{4} - {\sqrt{\frac{1 + 1 \frac{\sqrt{2}}{2}}{2}}}^{2}$

Step 1.7.4.3.2

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

$\frac{2 - \sqrt{2}}{4} - {\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}}^{2}$

Step 1.7.4.4

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

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

Step 1.7.4.5

Combine the numerators over the common denominator.

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

Step 1.7.4.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.7.4.7

Multiply $\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.7.4.7.1

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

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

Step 1.7.4.7.2

Multiply $2$ by $2$ .

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

Step 1.7.4.8

Rewrite $\sqrt{\frac{2 + \sqrt{2}}{4}}$ as $\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$ .

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

Step 1.7.4.9

Simplify the denominator.

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Step 1.7.4.9.1

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

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

Step 1.7.4.9.2

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

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

Step 1.8

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

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

Step 1.9

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

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Step 1.9.1

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

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

Step 1.9.2

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

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

Step 1.9.3

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

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

Step 1.9.4

Cancel the common factor of $2$ .

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Step 1.9.4.1

Cancel the common factor.

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

Step 1.9.4.2

Rewrite the expression.

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

Step 1.9.5

Simplify.

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

Step 1.10

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

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $- 1$ by $2$ .

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

Step 4

Simplify terms.

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Step 4.1

Subtract $2$ from $2$ .

$\frac{0 - \sqrt{2} - \sqrt{2}}{4}$

Step 4.2

Subtract $\sqrt{2}$ from $0$ .

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

Step 4.3

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

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $4$ .

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

Step 4.4.2.2

Cancel the common factor.

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

Step 4.4.2.3

Rewrite the expression.

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

Step 4.5

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.70710678 \dots$