Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Separate fractions.

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Write $\sin (x)$ as a fraction with denominator $1$ .

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

Step 1.6

Simplify.

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Step 1.6.1

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

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

Step 1.6.2

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

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

Step 1.7

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

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

Step 1.8

Simplify the numerator.

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Step 1.8.1

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

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

Step 1.8.2

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

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

Step 1.8.3

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

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

Step 1.8.4

Add $1$ and $1$ .

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

Step 1.9

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

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Step 1.9.1

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

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

Step 1.9.2

Multiply $\sin (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 1.9.2.1

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

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Step 1.9.2.1.1

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

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

Step 1.9.2.1.2

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

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

Step 1.9.2.2

Add $1$ and $2$ .

$\frac{\sin^{3} (x)}{\cos (x)} + \frac{\cot (x)}{\csc^{2} (x)}$

Step 1.10

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

$\frac{\sin^{3} (x)}{\cos (x)} + \frac{\cot (x)}{\csc (x) \csc (x)}$

Step 1.11

Separate fractions.

$\frac{\sin^{3} (x)}{\cos (x)} + \frac{1}{\csc (x)} \cdot \frac{\cot (x)}{\csc (x)}$

Step 1.12

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

$\frac{\sin^{3} (x)}{\cos (x)} + \frac{1}{\csc (x)} \cdot \frac{\cot (x)}{\frac{1}{\sin (x)}}$

Step 1.13

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

$\frac{\sin^{3} (x)}{\cos (x)} + \frac{1}{\csc (x)} \cdot \frac{\frac{\cos (x)}{\sin (x)}}{\frac{1}{\sin (x)}}$

Step 1.14

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$\frac{\sin^{3} (x)}{\cos (x)} + \frac{1}{\csc (x)} (\frac{\cos (x)}{\sin (x)} \sin (x))$

Step 1.15

Write $\sin (x)$ as a fraction with denominator $1$ .

$\frac{\sin^{3} (x)}{\cos (x)} + \frac{1}{\csc (x)} (\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{1})$

Step 1.16

Cancel the common factor of $\sin (x)$ .

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Step 1.16.1

Cancel the common factor.

$\frac{\sin^{3} (x)}{\cos (x)} + \frac{1}{\csc (x)} (\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{1})$

Step 1.16.2

Rewrite the expression.

$\frac{\sin^{3} (x)}{\cos (x)} + \frac{1}{\csc (x)} \cos (x)$

Step 1.17

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

$\frac{\sin^{3} (x)}{\cos (x)} + \frac{1}{\frac{1}{\sin (x)}} \cos (x)$

Step 1.18

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$\frac{\sin^{3} (x)}{\cos (x)} + 1 \sin (x) \cos (x)$

Step 1.19

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

$\frac{\sin^{3} (x)}{\cos (x)} + \sin (x) \cos (x)$

Step 2

Simplify each term.

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Step 2.1

Factor $\sin (x)$ out of $\sin^{3} (x)$ .

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

Step 2.2

Separate fractions.

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

Step 2.3

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

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

Step 2.4

Divide $\sin^{2} (x)$ by $1$ .

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