Algebra Examples

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

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

Factor $\sin (x)$ out of $\sin^{2} (x) - \sin (x) \cos (x)$ .

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Step 2.1.1

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

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

Step 2.1.2

Factor $\sin (x)$ out of $- \sin (x) \cos (x)$ .

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

Step 2.1.3

Factor $\sin (x)$ out of $\sin (x) \sin (x) + \sin (x) (- 1 \cos (x))$ .

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

Step 2.2

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

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

Step 3

Cancel the common factor of $\sin (x) - \cos (x)$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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