Find the Exact Value sin(225 degrees )^2-cos(225 degrees )^2

$\sin^{2} (225 °) - \cos^{2} (225 °)$

Step 1

Simplify each term.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$\frac{2}{2^{2}} - \cos^{2} (225 °)$

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

$\frac{2}{4} - \cos^{2} (225 °)$

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

Factor $2$ out of $2$ .

$\frac{2 (1)}{4} - \cos^{2} (225 °)$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{2} - \cos^{2} (225 °)$

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

Combine fractions.

Combine the numerators over the common denominator.

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

Simplify the expression.

Subtract $1$ from $1$ .

$\frac{0}{2}$

Divide $0$ by $2$ .

$0$

Find the Exact Value sin(225 degrees )^2-cos(225 degrees )^2

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