Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $\cot (2 x)$ in terms of sines and cosines.

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

Step 1.2

Rewrite $\tan (2 x)$ in terms of sines and cosines.

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

Step 2

Multiply the numerator and denominator of the fraction by $\sin (2 x) \cos (2 x)$ .

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Step 2.1

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

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

Step 2.2

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify by cancelling.

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Step 4.1

Cancel the common factor of $\sin (2 x)$ .

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Step 4.1.1

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

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 4.3

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

Cancel the common factor of $\cos (2 x)$ .

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Step 4.6.1

Move the leading negative in $- \frac{\sin (2 x)}{\cos (2 x)}$ into the numerator.

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

Step 4.6.2

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

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

Step 4.6.3

Cancel the common factor.

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

Step 4.6.4

Rewrite the expression.

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

Step 4.7

Raise $\sin (2 x)$ to the power of $1$ .

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

Step 4.8

Raise $\sin (2 x)$ to the power of $1$ .

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

Step 4.9

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

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

Step 4.10

Add $1$ and $1$ .

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

Step 5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (2 x)$ and $b = \sin (2 x)$ .

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

Step 6

Simplify the denominator.

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Step 6.1

Multiply $\sin (2 x) \cos (2 x) \sin (2 x)$ .

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Step 6.1.1

Raise $\sin (2 x)$ to the power of $1$ .

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

Step 6.1.2

Raise $\sin (2 x)$ to the power of $1$ .

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

Step 6.1.3

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

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

Step 6.1.4

Add $1$ and $1$ .

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

Step 6.2

Multiply $\sin (2 x) \cos (2 x) \cos (2 x)$ .

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Step 6.2.1

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 6.2.2

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 6.2.3

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

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

Step 6.2.4

Add $1$ and $1$ .

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

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

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

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

Step 6.3.3

Factor $\sin (2 x) \cos (2 x)$ out of $\sin (2 x) \cos (2 x) (\sin (2 x)) + \sin (2 x) \cos (2 x) (\cos (2 x))$ .

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