Trigonometry Examples

$\frac{3 a + 1}{a^{2} - 1} - \frac{1}{a + 1}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

$\frac{3 a + 1}{a^{2} - 1^{2}} - \frac{1}{a + 1}$

Step 1.2

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 $b = 1$ .

$\frac{3 a + 1}{(a + 1) (a - 1)} - \frac{1}{a + 1}$

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{1}{a + 1}$ by $\frac{a - 1}{a - 1}$ .

$\frac{3 a + 1}{(a + 1) (a - 1)} - \frac{a - 1}{(a + 1) (a - 1)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{3 a + 1 - (a - 1)}{(a + 1) (a - 1)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{3 a + 1 - a - - 1}{(a + 1) (a - 1)}$

Step 4.2

Multiply $- 1$ by $- 1$ .

$\frac{3 a + 1 - a + 1}{(a + 1) (a - 1)}$

Step 4.3

Subtract $a$ from $3 a$ .

$\frac{2 a + 1 + 1}{(a + 1) (a - 1)}$

Step 4.4

Add $1$ and $1$ .

$\frac{2 a + 2}{(a + 1) (a - 1)}$

Step 4.5

Factor $2$ out of $2 a + 2$ .

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Step 4.5.1

Factor $2$ out of $2 a$ .

$\frac{2 (a) + 2}{(a + 1) (a - 1)}$

Step 4.5.2

Factor $2$ out of $2$ .

$\frac{2 (a) + 2 (1)}{(a + 1) (a - 1)}$

Step 4.5.3

Factor $2$ out of $2 (a) + 2 (1)$ .

$\frac{2 (a + 1)}{(a + 1) (a - 1)}$

Step 5

Cancel the common factor of $a + 1$ .

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Step 5.1

Cancel the common factor.

$\frac{2 (a + 1)}{(a + 1) (a - 1)}$

Step 5.2

Rewrite the expression.

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