Trigonometry Examples

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

Step 1

Apply pythagorean identity.

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

Step 2

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

$\frac{\tan (x)}{\sin (x) \sin (x)}$

Step 3

Separate fractions.

$\frac{1}{\sin (x)} \cdot \frac{\tan (x)}{\sin (x)}$

Step 4

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 5

Rewrite $\frac{\frac{\sin (x)}{\cos (x)}}{\sin (x)}$ as a product.

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

Multiply $\frac{1}{\sin (x)}$ by $\frac{1}{\cos (x)}$ .

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

Step 8

Separate fractions.

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

Step 9

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{1}{\cos (x)} \csc (x)$

Step 10

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) \csc (x)$