Trigonometry Examples

$(1 - \sec (x)) (1 + \cos (x))$

Step 1

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

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

Step 2

Expand $(1 - \frac{1}{\cos (x)}) (1 + \cos (x))$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $1$ by $1$ .

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

Step 3.1.2

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

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

Step 3.1.3

Multiply $- 1$ by $1$ .

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

Step 3.1.4

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

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Step 3.1.4.1

Move the leading negative in $- \frac{1}{\cos (x)}$ into the numerator.

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

Step 3.1.4.2

Cancel the common factor.

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

Step 3.1.4.3

Rewrite the expression.

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

Step 3.2

Subtract $1$ from $1$ .

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

Step 3.3

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

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

Step 4

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\cos (x) - \sec (x)$