Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 1.5

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 1.6

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.6.1

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

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

Step 1.6.2

Cancel the common factors.

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

Step 2

Rearrange terms.

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

Step 3

Apply pythagorean identity.

$\csc^{2} (x)$