Algebra Examples

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

Step 1

Simplify with factoring out.

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Step 1.1

Reorder $\cos^{2} (x)$ and $- 1$ .

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

Step 1.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Rewrite $- 1 (1 - \cos^{2} (x))$ as $- (1 - \cos^{2} (x))$ .

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

Step 2

Apply pythagorean identity.

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

Step 3

Simplify with factoring out.

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Step 3.1

Reorder $\sin^{2} (x)$ and $- 1$ .

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

Step 3.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 3.3

Factor $- 1$ out of $\sin^{2} (x)$ .

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

Step 3.4

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (x))$ .

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

Step 3.5

Rewrite $- 1 (1 - \sin^{2} (x))$ as $- (1 - \sin^{2} (x))$ .

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

Step 4

Apply pythagorean identity.

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

Step 5

Dividing two negative values results in a positive value.

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

Step 6

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

$\tan^{2} (x)$