Precalculus Examples

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

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

Step 1

Expand

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

$(\sec (x) - 1) (\sec (x) + 1)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

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

Step 1.2

Apply the distributive property.

\sec (x) \sec (x) + \sec (x) \cdot 1 - 1 (\sec (x) + 1)

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

Step 1.3

Apply the distributive property.

\sec (x) \sec (x) + \sec (x) \cdot 1 - 1 \sec (x) - 1 \cdot 1

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

\sec (x) \sec (x) + \sec (x) \cdot 1 - 1 \sec (x) - 1 \cdot 1

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in

\sec (x) \sec (x) + \sec (x) \cdot 1 - 1 \sec (x) - 1 \cdot 1

$\sec (x) \sec (x) + \sec (x) \cdot 1 - 1 \sec (x) - 1 \cdot 1$ .

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Step 2.1.1

Reorder the factors in the terms

\sec (x) \cdot 1

$\sec (x) \cdot 1$ and

- 1 \sec (x)

$- 1 \sec (x)$ .

\sec (x) \sec (x) + 1 \sec (x) - 1 \sec (x) - 1 \cdot 1

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

Step 2.1.2

Subtract

1 \sec (x)

$1 \sec (x)$ from

1 \sec (x)

$1 \sec (x)$ .

\sec (x) \sec (x) + 0 - 1 \cdot 1

$\sec (x) \sec (x) + 0 - 1 \cdot 1$

Step 2.1.3

Add

\sec (x) \sec (x)

$\sec (x) \sec (x)$ and

0

$0$ .

\sec (x) \sec (x) - 1 \cdot 1

$\sec (x) \sec (x) - 1 \cdot 1$

\sec (x) \sec (x) - 1 \cdot 1

$\sec (x) \sec (x) - 1 \cdot 1$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply

\sec (x) \sec (x)

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

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Step 2.2.1.1

Raise

\sec (x)

$\sec (x)$ to the power of

1

$1$ .

\sec^{1} (x) \sec (x) - 1 \cdot 1

$\sec^{1} (x) \sec (x) - 1 \cdot 1$

Step 2.2.1.2

Raise

\sec (x)

$\sec (x)$ to the power of

1

$1$ .

\sec^{1} (x) \sec^{1} (x) - 1 \cdot 1

$\sec^{1} (x) \sec^{1} (x) - 1 \cdot 1$

Step 2.2.1.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

{\sec (x)}^{1 + 1} - 1 \cdot 1

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

Step 2.2.1.4

Add

1

$1$ and

1

$1$ .

\sec^{2} (x) - 1 \cdot 1

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

\sec^{2} (x) - 1 \cdot 1

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

Step 2.2.2

Multiply

- 1

$- 1$ by

1

$1$ .

\sec^{2} (x) - 1

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

\sec^{2} (x) - 1

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

\sec^{2} (x) - 1

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

Step 3

Apply pythagorean identity.

\tan^{2} (x)

$\tan^{2} (x)$