Trigonometry Examples

$\cos (\frac{π}{4}) \cos (\frac{π}{4}) - \sin (\frac{π}{4}) \sin (\frac{π}{4})$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} \cos (\frac{π}{4}) - \sin (\frac{π}{4}) \sin (\frac{π}{4})$

Step 1.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (\frac{π}{4}) \sin (\frac{π}{4})$

Step 1.3

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

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Step 1.3.1

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

$\frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} - \sin (\frac{π}{4}) \sin (\frac{π}{4})$

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Add $1$ and $1$ .

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

Step 1.3.6

Multiply $2$ by $2$ .

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

$\frac{2^{\frac{1}{2} \cdot 2}}{4} - \sin (\frac{π}{4}) \sin (\frac{π}{4})$

Step 1.4.3

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

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

Step 1.4.4

Cancel the common factor of $2$ .

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Step 1.4.4.1

Cancel the common factor.

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

Step 1.4.4.2

Rewrite the expression.

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

Step 1.4.5

Evaluate the exponent.

$\frac{2}{4} - \sin (\frac{π}{4}) \sin (\frac{π}{4})$

Step 1.5

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

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Step 1.5.1

Factor $2$ out of $2$ .

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

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2} - \sin (\frac{π}{4}) \sin (\frac{π}{4})$

Step 1.5.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2} - \sin (\frac{π}{4}) \sin (\frac{π}{4})$

Step 1.5.2.3

Rewrite the expression.

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

Step 1.6

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{1}{2} - \frac{\sqrt{2}}{2} \sin (\frac{π}{4})$

Step 1.7

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 1.8

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

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Step 1.8.1

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

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

Step 1.8.2

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

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

Step 1.8.3

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

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

Step 1.8.4

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

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

Step 1.8.5

Add $1$ and $1$ .

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

Step 1.8.6

Multiply $2$ by $2$ .

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

Step 1.9

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

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Step 1.9.1

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

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

Step 1.9.2

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

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

Step 1.9.3

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

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

Step 1.9.4

Cancel the common factor of $2$ .

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Step 1.9.4.1

Cancel the common factor.

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

Step 1.9.4.2

Rewrite the expression.

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

Step 1.9.5

Evaluate the exponent.

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

Step 1.10

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

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Step 1.10.1

Factor $2$ out of $2$ .

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

Step 1.10.2

Cancel the common factors.

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Step 1.10.2.1

Factor $2$ out of $4$ .

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

Step 1.10.2.2

Cancel the common factor.

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

Step 1.10.2.3

Rewrite the expression.

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

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Simplify the expression.

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Step 2.2.1

Subtract $1$ from $1$ .

$\frac{0}{2}$

Step 2.2.2

Divide $0$ by $2$ .

$0$