Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Combine $- \cot (x)$ and $\frac{1 - \tan (x)}{1 - \tan (x)}$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $- 1$ by $1$ .

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

Step 3.3

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

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Step 3.3.1

Add parentheses.

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

Step 3.3.2

Reorder $\cot (x)$ and $- \tan (x)$ .

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

Step 3.3.3

Add parentheses.

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

Step 3.3.4

Reorder $\tan (x)$ and $\cot (x)$ .

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

Step 3.3.5

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

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

Step 3.3.6

Cancel the common factors.

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

Step 3.4

Multiply $- (- 1 \cdot 1)$ .

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Step 3.4.1

Multiply $- 1$ by $1$ .

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

Step 3.4.2

Multiply $- 1$ by $- 1$ .

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

Step 3.5

Subtract $\cot (x)$ from $\cot (x)$ .

$\frac{0 - \tan (x) + 1}{1 - \tan (x)}$

Step 3.6

Subtract $\tan (x)$ from $0$ .

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

Step 4

Cancel the common factor of $- \tan (x) + 1$ and $1 - \tan (x)$ .

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Step 4.1

Reorder terms.

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

$1$