Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

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

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Step 1.4.1.1.1

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

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

Step 1.4.1.1.2

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

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

Step 1.4.1.1.3

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

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

Step 1.4.1.1.4

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

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

Step 1.4.1.1.5

Add $1$ and $1$ .

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

Step 1.4.1.2

Rewrite using the commutative property of multiplication.

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

Step 1.4.1.3

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

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Step 1.4.1.3.1

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

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

Step 1.4.1.3.2

Cancel the common factor.

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

Step 1.4.1.3.3

Rewrite the expression.

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

Step 1.4.1.4

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

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Step 1.4.1.4.1

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

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

Step 1.4.1.4.2

Cancel the common factor.

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

Step 1.4.1.4.3

Rewrite the expression.

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

Step 1.4.1.5

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

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Step 1.4.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 1.4.1.5.2

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

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

Step 1.4.1.5.3

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

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

Step 1.4.1.5.4

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

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

Step 1.4.1.5.5

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

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

Step 1.4.1.5.6

Add $1$ and $1$ .

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

Step 1.4.2

Subtract $1$ from $- 1$ .

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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Step 1.7.1

Apply the distributive property.

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

Step 1.7.2

Apply the distributive property.

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

Step 1.7.3

Apply the distributive property.

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

Step 1.8

Simplify and combine like terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

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

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Step 1.8.1.1.1

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

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

Step 1.8.1.1.2

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

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

Step 1.8.1.1.3

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

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

Step 1.8.1.1.4

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

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

Step 1.8.1.1.5

Add $1$ and $1$ .

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

Step 1.8.1.2

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

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Step 1.8.1.2.1

Cancel the common factor.

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

Step 1.8.1.2.2

Rewrite the expression.

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

Step 1.8.1.3

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

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Step 1.8.1.3.1

Cancel the common factor.

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

Step 1.8.1.3.2

Rewrite the expression.

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

Step 1.8.1.4

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

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Step 1.8.1.4.1

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

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

Step 1.8.1.4.2

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

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

Step 1.8.1.4.3

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

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

Step 1.8.1.4.4

Add $1$ and $1$ .

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

Step 1.8.2

Add $1$ and $1$ .

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

Step 1.9

Apply the distributive property.

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

Step 1.10

Multiply $- 1$ by $2$ .

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

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $\frac{1}{\cos^{2} (x)} - 2 + \cos^{2} (x) - \frac{1}{\cos^{2} (x)} - 2 - \cos^{2} (x)$ .

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Step 2.1.1

Subtract $\frac{1}{\cos^{2} (x)}$ from $\frac{1}{\cos^{2} (x)}$ .

$0 - 2 + \cos^{2} (x) - 2 - \cos^{2} (x)$

Step 2.1.2

Subtract $2$ from $0$ .

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

Step 2.1.3

Subtract $\cos^{2} (x)$ from $\cos^{2} (x)$ .

$- 2 + 0 - 2$

Step 2.1.4

Add $- 2$ and $0$ .

$- 2 - 2$

Step 2.2

Subtract $2$ from $- 2$ .

$- 4$