Precalculus Examples

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

Step 1

To write $\frac{\cot (x)}{1 - \tan (x)}$ as a fraction with a common denominator, multiply by $\frac{1 - \cot (x)}{1 - \cot (x)}$ .

$\frac{\cot (x)}{1 - \tan (x)} \cdot \frac{1 - \cot (x)}{1 - \cot (x)} + \frac{\tan (x)}{1 - \cot (x)}$

Step 2

To write $\frac{\tan (x)}{1 - \cot (x)}$ as a fraction with a common denominator, multiply by $\frac{1 - \tan (x)}{1 - \tan (x)}$ .

$\frac{\cot (x)}{1 - \tan (x)} \cdot \frac{1 - \cot (x)}{1 - \cot (x)} + \frac{\tan (x)}{1 - \cot (x)} \cdot \frac{1 - \tan (x)}{1 - \tan (x)}$

Step 3

Write each expression with a common denominator of $(1 - \tan (x)) (1 - \cot (x))$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{\cot (x)}{1 - \tan (x)}$ by $\frac{1 - \cot (x)}{1 - \cot (x)}$ .

$\frac{\cot (x) (1 - \cot (x))}{(1 - \tan (x)) (1 - \cot (x))} + \frac{\tan (x)}{1 - \cot (x)} \cdot \frac{1 - \tan (x)}{1 - \tan (x)}$

Step 3.2

Multiply $\frac{\tan (x)}{1 - \cot (x)}$ by $\frac{1 - \tan (x)}{1 - \tan (x)}$ .

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

Step 3.3

Reorder the factors of $(1 - \tan (x)) (1 - \cot (x))$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{\cot (x) \cdot 1 + \cot (x) (- \cot (x)) + \tan (x) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))}$

Step 5.2

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

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

Step 5.3

Rewrite using the commutative property of multiplication.

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

Step 5.4

Multiply $- \cot (x) \cot (x)$ .

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Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.4.4

Add $1$ and $1$ .

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

Step 5.5

Apply the distributive property.

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

Step 5.6

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

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

Step 5.7

Rewrite using the commutative property of multiplication.

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

Step 5.8

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

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Step 5.8.1

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

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

Step 5.8.2

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

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

Step 5.8.3

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

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

Step 5.8.4

Add $1$ and $1$ .

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