Algebra Examples

$3 (\cos^{2} (45 °) - \sin^{2} (90 °))$

Step 1

Simplify each term.

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Step 1.1

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

$3 ({(\frac{\sqrt{2}}{2})}^{2} - \sin^{2} (90 °))$

Step 1.2

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

$3 (\frac{{\sqrt{2}}^{2}}{2^{2}} - \sin^{2} (90 °))$

Step 1.3

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

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Step 1.3.1

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

$3 (\frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} - \sin^{2} (90 °))$

Step 1.3.2

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

$3 (\frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} - \sin^{2} (90 °))$

Step 1.3.3

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

$3 (\frac{2^{\frac{2}{2}}}{2^{2}} - \sin^{2} (90 °))$

Step 1.3.4

Cancel the common factor of $2$ .

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Step 1.3.4.1

Cancel the common factor.

$3 (\frac{2^{\frac{2}{2}}}{2^{2}} - \sin^{2} (90 °))$

Step 1.3.4.2

Rewrite the expression.

$3 (\frac{2^{1}}{2^{2}} - \sin^{2} (90 °))$

Step 1.3.5

Evaluate the exponent.

$3 (\frac{2}{2^{2}} - \sin^{2} (90 °))$

Step 1.4

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

$3 (\frac{2}{4} - \sin^{2} (90 °))$

Step 1.5

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

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Step 1.5.1

Factor $2$ out of $2$ .

$3 (\frac{2 (1)}{4} - \sin^{2} (90 °))$

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $2$ out of $4$ .

$3 (\frac{2 \cdot 1}{2 \cdot 2} - \sin^{2} (90 °))$

Step 1.5.2.2

Cancel the common factor.

$3 (\frac{2 \cdot 1}{2 \cdot 2} - \sin^{2} (90 °))$

Step 1.5.2.3

Rewrite the expression.

$3 (\frac{1}{2} - \sin^{2} (90 °))$

Step 1.6

The exact value of $\sin (90 °)$ is $1$ .

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

Step 1.7

One to any power is one.

$3 (\frac{1}{2} - 1 \cdot 1)$

Step 1.8

Multiply $- 1$ by $1$ .

$3 (\frac{1}{2} - 1)$

Step 2

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

$3 (\frac{1}{2} - 1 \cdot \frac{2}{2})$

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

$3 \frac{1 - 1 \cdot 2}{2}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $2$ .

$3 \frac{1 - 2}{2}$

Step 5.2

Subtract $2$ from $1$ .

$3 (\frac{- 1}{2})$

Step 6

Move the negative in front of the fraction.

$3 (- \frac{1}{2})$

Step 7

Multiply $3 (- \frac{1}{2})$ .

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Step 7.1

Multiply $- 1$ by $3$ .

$- 3 (\frac{1}{2})$

Step 7.2

Combine $- 3$ and $\frac{1}{2}$ .

$\frac{- 3}{2}$

Step 8

Move the negative in front of the fraction.

$- \frac{3}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{2}$

Decimal Form:

$- 1.5$

Mixed Number Form:

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