Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify terms.

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Step 4.1

Combine the opposite terms in $\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} (- \cos (x)) + \cos (x) \frac{1}{\cos (x)} + \cos (x) (- \cos (x))$ .

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Step 4.1.1

Reorder the factors in the terms $\frac{1}{\cos (x)} (- \cos (x))$ and $\cos (x) \frac{1}{\cos (x)}$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

Simplify each term.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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

Step 4.2.1.4

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

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

Step 4.2.1.5

Add $1$ and $1$ .

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

Step 4.2.2

Rewrite using the commutative property of multiplication.

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

Step 4.2.3

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

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Step 4.2.3.1

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

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

Step 4.2.3.2

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

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

Step 4.2.3.3

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

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

Step 4.2.3.4

Add $1$ and $1$ .

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

Step 4.3

Rewrite $1$ as $1^{2}$ .

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

Step 5

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

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

Step 6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{1}{\cos (x)}$ and $b = \cos (x)$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Expand $(\sec (x) + \cos (x)) (\sec (x) - \cos (x))$ using the FOIL Method.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Apply the distributive property.

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

Step 10

Simplify terms.

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Step 10.1

Combine the opposite terms in $\sec (x) \sec (x) + \sec (x) (- \cos (x)) + \cos (x) \sec (x) + \cos (x) (- \cos (x))$ .

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Step 10.1.1

Reorder the factors in the terms $\sec (x) (- \cos (x))$ and $\cos (x) \sec (x)$ .

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

Step 10.1.2

Add $- \cos (x) \sec (x)$ and $\cos (x) \sec (x)$ .

$\sec (x) \sec (x) + 0 + \cos (x) (- \cos (x))$

Step 10.1.3

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

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

Step 10.2

Simplify each term.

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Step 10.2.1

Multiply $\sec (x) \sec (x)$ .

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Step 10.2.1.1

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

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

Step 10.2.1.2

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

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

Step 10.2.1.3

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

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

Step 10.2.1.4

Add $1$ and $1$ .

$\sec^{2} (x) + \cos (x) (- \cos (x))$

Step 10.2.2

Rewrite using the commutative property of multiplication.

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

Step 10.2.3

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

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Step 10.2.3.1

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

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

Step 10.2.3.2

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

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

Step 10.2.3.3

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

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

Step 10.2.3.4

Add $1$ and $1$ .

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