Trigonometry Examples

$(1 - \tan (x)) (1 + \cot (x))$

Step 1

Expand $(1 - \tan (x)) (1 + \cot (x))$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$1 (1 + \cot (x)) - \tan (x) (1 + \cot (x))$

Step 1.2

Apply the distributive property.

$1 \cdot 1 + 1 \cot (x) - \tan (x) (1 + \cot (x))$

Step 1.3

Apply the distributive property.

$1 \cdot 1 + 1 \cot (x) - \tan (x) \cdot 1 - \tan (x) \cot (x)$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $1$ by $1$ .

$1 + 1 \cot (x) - \tan (x) \cdot 1 - \tan (x) \cot (x)$

Step 2.1.2

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

$1 + \cot (x) - \tan (x) \cdot 1 - \tan (x) \cot (x)$

Step 2.1.3

Multiply $- 1$ by $1$ .

$1 + \cot (x) - \tan (x) - \tan (x) \cot (x)$

Step 2.1.4

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

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Step 2.1.4.1

Add parentheses.

$1 + \cot (x) - \tan (x) - (\tan (x) \cot (x))$

Step 2.1.4.2

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

$1 + \cot (x) - \tan (x) - (\cot (x) \tan (x))$

Step 2.1.4.3

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

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

Step 2.1.4.4

Cancel the common factors.

$1 + \cot (x) - \tan (x) - 1 \cdot 1$

Step 2.1.5

Multiply $- 1$ by $1$ .

$1 + \cot (x) - \tan (x) - 1$

Step 2.2

Subtract $1$ from $1$ .

$0 + \cot (x) - \tan (x)$

Step 2.3

Add $0$ and $\cot (x)$ .

$\cot (x) - \tan (x)$