Trigonometry Examples

${(\frac{5}{13})}^{2} - {(- \frac{12}{13})}^{2}$

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $\frac{5}{13}$ .

$\frac{5^{2}}{13^{2}} - {(- \frac{12}{13})}^{2}$

Step 1.2

Raise $5$ to the power of $2$ .

$\frac{25}{13^{2}} - {(- \frac{12}{13})}^{2}$

Step 1.3

Raise $13$ to the power of $2$ .

$\frac{25}{169} - {(- \frac{12}{13})}^{2}$

Step 1.4

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

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Step 1.4.1

Apply the product rule to $- \frac{12}{13}$ .

$\frac{25}{169} - ({(- 1)}^{2} {(\frac{12}{13})}^{2})$

Step 1.4.2

Apply the product rule to $\frac{12}{13}$ .

$\frac{25}{169} - ({(- 1)}^{2} \frac{12^{2}}{13^{2}})$

Step 1.5

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

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Step 1.5.1

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

$\frac{25}{169} + {(- 1)}^{2} \cdot - 1 \frac{12^{2}}{13^{2}}$

Step 1.5.2

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

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Step 1.5.2.1

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

$\frac{25}{169} + {(- 1)}^{2} \cdot {(- 1)}^{1} \frac{12^{2}}{13^{2}}$

Step 1.5.2.2

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

$\frac{25}{169} + {(- 1)}^{2 + 1} \frac{12^{2}}{13^{2}}$

Step 1.5.3

Add $2$ and $1$ .

$\frac{25}{169} + {(- 1)}^{3} \frac{12^{2}}{13^{2}}$

Step 1.6

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

$\frac{25}{169} - \frac{12^{2}}{13^{2}}$

Step 1.7

Raise $12$ to the power of $2$ .

$\frac{25}{169} - \frac{144}{13^{2}}$

Step 1.8

Raise $13$ to the power of $2$ .

$\frac{25}{169} - \frac{144}{169}$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{25 - 144}{169}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Subtract $144$ from $25$ .

$\frac{- 119}{169}$

Step 2.2.2

Move the negative in front of the fraction.

$- \frac{119}{169}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \frac{119}{169}$

Decimal Form:

$- 0.70414201 \dots$