Trigonometry Examples

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

Step 1

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

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

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.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $1$ .

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

Step 3.1.2

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

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

Step 3.1.3

Multiply $- 1$ by $1$ .

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

Step 3.1.4

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

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Step 3.1.4.1

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

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

Step 3.1.4.2

Cancel the common factor.

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

Step 3.1.4.3

Rewrite the expression.

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

Step 3.2

Subtract $1$ from $1$ .

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

Step 3.3

Add $0$ and $\frac{1}{\cos (x)}$ .

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

Step 4

Apply the distributive property.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Add $1$ and $1$ .

$1 - \cos^{2} (x)$

Step 7

Apply pythagorean identity.

$\sin^{2} (x)$