Simplify ((c^2-1)/(c^4-1))÷((c+1)/(c^2+1))

$\frac{c^{2} - 1}{c^{4} - 1} \div \frac{c + 1}{c^{2} + 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{c^{2} - 1}{c^{4} - 1} \cdot \frac{c^{2} + 1}{c + 1}$

Step 2

Simplify the numerator.

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

$\frac{c^{2} - 1^{2}}{c^{4} - 1} \cdot \frac{c^{2} + 1}{c + 1}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = c$ and $b = 1$ .

$\frac{(c + 1) (c - 1)}{c^{4} - 1} \cdot \frac{c^{2} + 1}{c + 1}$

Step 3

Simplify the denominator.

Rewrite $c^{4}$ as ${(c^{2})}^{2}$ .

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

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

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = c^{2}$ and $b = 1$ .

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

Simplify.

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

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = c$ and $b = 1$ .

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

Step 4

Reduce the expression by cancelling the common factors.

Cancel the common factor of $c + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $c^{2} + 1$ .

Factor $c^{2} + 1$ out of $(c^{2} + 1) (c + 1) (c - 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{c - 1}{(c + 1) (c - 1)}$

Cancel the common factor of $c - 1$ .

Cancel the common factor.

$\frac{c - 1}{(c + 1) (c - 1)}$

Rewrite the expression.

$\frac{1}{c + 1}$

Simplify ((c^2-1)/(c^4-1))÷((c+1)/(c^2+1))

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