Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Separate fractions.

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify each term.

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Step 5.1

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

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

Step 5.2

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

$(\tan (x) + \cot (x)) \tan (x)$

Step 6

Apply the distributive property.

$\tan (x) \tan (x) + \cot (x) \tan (x)$

Step 7

Multiply $\tan (x) \tan (x)$ .

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Step 7.1

Raise $\tan (x)$ to the power of $1$ .

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

Step 7.2

Raise $\tan (x)$ to the power of $1$ .

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

Step 7.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.4

Add $1$ and $1$ .

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Cancel the common factors.

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

Step 9

Apply pythagorean identity.

$\sec^{2} (x)$