Trigonometry Examples

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

Step 1

Factor $\cot^{2} (x)$ out of $\cot^{2} (x) \tan^{2} (x) + \cot^{2} (x)$ .

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Step 1.1

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

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

Step 1.2

Multiply by $1$ .

$\cot^{2} (x) (\tan^{2} (x)) + \cot^{2} (x) \cdot 1$

Step 1.3

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

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

Step 2

Apply pythagorean identity.

$\cot^{2} (x) \sec^{2} (x)$

Step 3

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

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

Step 4

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

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

Step 5

Simplify the expression.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

One to any power is one.

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

Step 6

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

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

Step 7

Cancel the common factor of $\cos^{2} (x)$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

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

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

Step 9

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

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

Step 10

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

$\csc^{2} (x)$