Algebra Examples

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

Step 1

Simplify with factoring out.

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Step 1.1

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

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

Step 1.2

Factor $\cot (x)$ out of $- \cot (x)$ .

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

Step 1.3

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

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

Step 2

Apply pythagorean identity.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

Add $1$ and $1$ .

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

Step 5

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

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

Step 6

Simplify the numerator.

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Step 6.1

Reduce the expression $\frac{\cos (x) \sin^{2} (x)}{\sin (x) \cos^{2} (x)}$ by cancelling the common factors.

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Step 6.1.1

Factor $\cos (x)$ out of $\cos (x) \sin^{2} (x)$ .

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

Step 6.1.2

Factor $\cos (x)$ out of $\sin (x) \cos^{2} (x)$ .

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

Step 6.1.3

Cancel the common factor.

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

Step 6.1.4

Rewrite the expression.

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $\sin (x)$ out of $\sin (x) \cos (x)$ .

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

Step 6.2.2.2

Cancel the common factor.

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

Step 6.2.2.3

Rewrite the expression.

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

Step 7

Multiply the numerator by the reciprocal of the denominator.

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

Step 8

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

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

Step 9

Separate fractions.

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

Step 10

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

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

Step 11

Simplify each term.

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Step 11.1

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

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

Step 11.2

Separate fractions.

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

Step 11.3

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

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

Step 11.4

Divide $\sin (x)$ by $1$ .

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

Step 12

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

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