Simplify (sin(x))/(cos(x)-cos(x)^3)

$\frac{\sin (x)}{\cos (x) - \cos^{3} (x)}$

Step 1

Simplify the denominator.

Factor $\cos (x)$ out of $\cos (x) - \cos^{3} (x)$ .

Multiply by $1$ .

$\frac{\sin (x)}{\cos (x) \cdot 1 - \cos^{3} (x)}$

Factor $\cos (x)$ out of $- \cos^{3} (x)$ .

$\frac{\sin (x)}{\cos (x) \cdot 1 + \cos (x) (- \cos^{2} (x))}$

Factor $\cos (x)$ out of $\cos (x) \cdot 1 + \cos (x) (- \cos^{2} (x))$ .

$\frac{\sin (x)}{\cos (x) (1 - \cos^{2} (x))}$

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

$\frac{\sin (x)}{\cos (x) (1^{2} - \cos^{2} (x))}$

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

$\frac{\sin (x)}{\cos (x) (1 + \cos (x)) (1 - \cos (x))}$

Step 2

Separate fractions.

$\frac{1}{(1 + \cos (x)) (1 - \cos (x))} \cdot \frac{\sin (x)}{\cos (x)}$

Step 3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{1}{(1 + \cos (x)) (1 - \cos (x))} \tan (x)$

Step 4

Combine $\frac{1}{(1 + \cos (x)) (1 - \cos (x))}$ and $\tan (x)$ .

$\frac{\tan (x)}{(1 + \cos (x)) (1 - \cos (x))}$

Simplify (sin(x))/(cos(x)-cos(x)^3)

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