Trigonometry Examples

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

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

Step 1

Simplify each term.

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Step 1.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

Step 1.2

Rewrite

\frac{1^{2}}{\tan^{2} (x)}

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

{(\frac{1}{\tan (x)})}^{2}

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

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

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

Step 1.3

Rewrite

\tan (x)

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

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

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

Step 1.4

Multiply by the reciprocal of the fraction to divide by

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

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

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

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

Step 1.5

Convert from

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

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

\cot (x)

$\cot (x)$ .

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

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

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

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

Step 1.6.2

Cancel the common factors.

\cot^{2} (x) + 1

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

\cot^{2} (x) + 1

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

\cot^{2} (x) + 1

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

Step 2

Rearrange terms.

1 + \cot^{2} (x)

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

Step 3

Apply pythagorean identity.

\csc^{2} (x)

$\csc^{2} (x)$