Trigonometry Examples

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

Step 1

Factor $1$ out of $\frac{\cot^{2} (x)}{1 - \sin^{2} (x)} + \frac{\tan^{2} (x)}{1 - \cos^{2} (x)}$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Factor $1$ out of $1 \frac{\cot^{2} (x)}{1 - \sin^{2} (x)} + 1 \frac{\tan^{2} (x)}{1 - \cos^{2} (x)}$ .

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

Step 2

Apply pythagorean identity.

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

Step 3

Apply pythagorean identity.

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

Step 4

Simplify terms.

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Step 4.1

Multiply $\frac{\cot^{2} (x)}{\cos^{2} (x)} + \frac{\tan^{2} (x)}{\sin^{2} (x)}$ by $1$ .

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

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite $\frac{\cot^{2} (x)}{\cos^{2} (x)}$ as ${(\frac{\cot (x)}{\cos (x)})}^{2}$ .

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

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

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Step 4.2.4.1

Cancel the common factor.

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

Step 4.2.4.2

Rewrite the expression.

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

Step 4.2.5

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 4.2.6

One to any power is one.

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

Step 4.2.7

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

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

Step 4.2.8

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

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

Step 4.2.9

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

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

Step 4.2.10

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

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Step 4.2.10.1

Cancel the common factor.

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

Step 4.2.10.2

Rewrite the expression.

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

Step 4.2.11

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 4.2.12

One to any power is one.

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

Step 4.3

Simplify each term.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

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

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

Step 4.3.6

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

$\csc^{2} (x) + \sec^{2} (x)$