Trigonometry Examples

$\frac{\frac{5}{a + b} + \frac{5}{a - b}}{\frac{10}{a^{2} - b^{2}}}$

Step 1

To write $\frac{5}{a + b}$ as a fraction with a common denominator, multiply by $\frac{a - b}{a - b}$ .

$\frac{\frac{5}{a + b} \cdot \frac{a - b}{a - b} + \frac{5}{a - b}}{\frac{10}{a^{2} - b^{2}}}$

Step 2

To write $\frac{5}{a - b}$ as a fraction with a common denominator, multiply by $\frac{a + b}{a + b}$ .

$\frac{\frac{5}{a + b} \cdot \frac{a - b}{a - b} + \frac{5}{a - b} \cdot \frac{a + b}{a + b}}{\frac{10}{a^{2} - b^{2}}}$

Step 3

Write each expression with a common denominator of $(a + b) (a - b)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5}{a + b}$ by $\frac{a - b}{a - b}$ .

$\frac{\frac{5 (a - b)}{(a + b) (a - b)} + \frac{5}{a - b} \cdot \frac{a + b}{a + b}}{\frac{10}{a^{2} - b^{2}}}$

Step 3.2

Multiply $\frac{5}{a - b}$ by $\frac{a + b}{a + b}$ .

$\frac{\frac{5 (a - b)}{(a + b) (a - b)} + \frac{5 (a + b)}{(a - b) (a + b)}}{\frac{10}{a^{2} - b^{2}}}$

Step 3.3

Reorder the factors of $(a - b) (a + b)$ .

$\frac{\frac{5 (a - b)}{(a + b) (a - b)} + \frac{5 (a + b)}{(a + b) (a - b)}}{\frac{10}{a^{2} - b^{2}}}$

Step 4

Combine the numerators over the common denominator.

$\frac{\frac{5 (a - b) + 5 (a + b)}{(a + b) (a - b)}}{\frac{10}{a^{2} - b^{2}}}$

Step 5

Rewrite $\frac{5 (a - b) + 5 (a + b)}{(a + b) (a - b)}$ in a factored form.

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Step 5.1

Factor $5$ out of $5 (a - b) + 5 (a + b)$ .

$\frac{\frac{5 (a - b + a + b)}{(a + b) (a - b)}}{\frac{10}{a^{2} - b^{2}}}$

Step 5.2

Add $a$ and $a$ .

$\frac{\frac{5 (2 a - b + b)}{(a + b) (a - b)}}{\frac{10}{a^{2} - b^{2}}}$

Step 5.3

Add $- b$ and $b$ .

$\frac{\frac{5 (2 a + 0)}{(a + b) (a - b)}}{\frac{10}{a^{2} - b^{2}}}$

Step 5.4

Factor.

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Step 5.4.1

Add $2 a$ and $0$ .

$\frac{\frac{5 (2 a)}{(a + b) (a - b)}}{\frac{10}{a^{2} - b^{2}}}$

Step 5.4.2

Remove unnecessary parentheses.

$\frac{\frac{5 \cdot 2 a}{(a + b) (a - b)}}{\frac{10}{a^{2} - b^{2}}}$

Step 5.5

Multiply $5$ by $2$ .

$\frac{\frac{10 a}{(a + b) (a - b)}}{\frac{10}{a^{2} - b^{2}}}$

Step 6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = b$ .

$\frac{\frac{10 a}{(a + b) (a - b)}}{\frac{10}{(a + b) (a - b)}}$