Trigonometry Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Apply the distributive property.

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

Step 1.1.3

Apply the distributive property.

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

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

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

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Step 1.2.1.1.1

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

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

Step 1.2.1.1.2

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

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

Step 1.2.1.1.3

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

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

Step 1.2.1.1.4

Add $1$ and $1$ .

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

Step 1.2.1.2

Move $- 1$ to the left of $\sec (x)$ .

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

Step 1.2.1.3

Rewrite $- 1 \sec (x)$ as $- \sec (x)$ .

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

Step 1.2.1.4

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

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

Step 1.2.1.5

Multiply $- 1$ by $1$ .

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

Step 1.2.2

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

$\sec^{2} (x) + 0 - 1 = \tan^{2} (x)$

Step 1.2.3

Add $\sec^{2} (x)$ and $0$ .

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

Step 1.3

Apply pythagorean identity.

$\tan^{2} (x) = \tan^{2} (x)$

Step 2

For the two functions to be equal, the arguments of each must be equal.

$x = x$

Step 3

Move all terms containing $x$ to the left side of the equation.

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Step 3.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 3.2

Subtract $x$ from $x$ .

$0 = 0$

Step 4

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$