Trigonometry Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $1$ .

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.1.5

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

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Step 3.1.5.1

Factor $\cos (x)$ out of $- \cos (x)$ .

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

Step 3.1.5.2

Cancel the common factor.

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

Step 3.1.5.3

Rewrite the expression.

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

Step 3.2

Subtract $1$ from $1$ .

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

Step 3.3

Subtract $\frac{1}{\cos (x)}$ from $0$ .

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

Step 4

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

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