Precalculus Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $\sec (x) \sec (x) + \sec (x) \cdot 1 - 1 \sec (x) - 1 \cdot 1$ .

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Step 2.1.1

Reorder the factors in the terms $\sec (x) \cdot 1$ and $- 1 \sec (x)$ .

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

Step 2.1.2

Subtract $1 \sec (x)$ from $1 \sec (x)$ .

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

Step 2.1.3

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

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Add $1$ and $1$ .

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

Step 2.2.2

Multiply $- 1$ by $1$ .

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

Step 3

Apply pythagorean identity.

$\tan^{2} (x)$