Algebra Examples

$\sin^{2} (225 °) - \cos^{2} (225 °)$

Step 1

Simplify each term.

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Step 1.1

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

${(- \sin (45))}^{2} - \cos^{2} (225 °)$

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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Step 1.6.1

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

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

Step 1.6.2

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

$\frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} - \cos^{2} (225 °)$

Step 1.6.3

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

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

Step 1.6.4

Cancel the common factor of $2$ .

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Step 1.6.4.1

Cancel the common factor.

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

Step 1.6.4.2

Rewrite the expression.

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

Step 1.6.5

Evaluate the exponent.

$\frac{2}{2^{2}} - \cos^{2} (225 °)$

Step 1.7

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

$\frac{2}{4} - \cos^{2} (225 °)$

Step 1.8

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

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Step 1.8.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4} - \cos^{2} (225 °)$

Step 1.8.2

Cancel the common factors.

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Step 1.8.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2} - \cos^{2} (225 °)$

Step 1.8.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2} - \cos^{2} (225 °)$

Step 1.8.2.3

Rewrite the expression.

$\frac{1}{2} - \cos^{2} (225 °)$

Step 1.9

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.

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

Step 1.10

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

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

Step 1.11

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

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Step 1.11.1

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

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

Step 1.11.2

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

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

Step 1.12

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.12.1

Move ${(- 1)}^{2}$ .

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

Step 1.12.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 1.12.2.1

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

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

Step 1.12.2.2

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

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

Step 1.12.3

Add $2$ and $1$ .

$\frac{1}{2} + {(- 1)}^{3} \frac{{\sqrt{2}}^{2}}{2^{2}}$

Step 1.13

Raise $- 1$ to the power of $3$ .

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

Step 1.14

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

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Step 1.14.1

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

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

Step 1.14.2

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

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

Step 1.14.3

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

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

Step 1.14.4

Cancel the common factor of $2$ .

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Step 1.14.4.1

Cancel the common factor.

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

Step 1.14.4.2

Rewrite the expression.

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

Step 1.14.5

Evaluate the exponent.

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

Step 1.15

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

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

Step 1.16

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

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Step 1.16.1

Factor $2$ out of $2$ .

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

Step 1.16.2

Cancel the common factors.

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Step 1.16.2.1

Factor $2$ out of $4$ .

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

Step 1.16.2.2

Cancel the common factor.

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

Step 1.16.2.3

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 $2$ .

$0$