Algebra Examples

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

Step 1

Rewrite $\sec^{2} (x) - \tan^{2} (x)$ as a difference of squares.

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

Step 2

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

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

Step 3

Simplify the left side.

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Step 3.1

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

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Step 3.1.1

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

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Step 3.1.1.1

Apply the distributive property.

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

Step 3.1.1.2

Apply the distributive property.

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

Step 3.1.1.3

Apply the distributive property.

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

Step 3.1.2

Simplify terms.

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Step 3.1.2.1

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

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Step 3.1.2.1.1

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

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

Step 3.1.2.1.2

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

$\sec (x) \sec (x) + 0 + \tan (x) (- \tan (x)) = 1$

Step 3.1.2.1.3

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

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

Step 3.1.2.2

Simplify each term.

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Step 3.1.2.2.1

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

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Step 3.1.2.2.1.1

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

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

Step 3.1.2.2.1.2

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

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

Step 3.1.2.2.1.3

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

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

Step 3.1.2.2.1.4

Add $1$ and $1$ .

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

Step 3.1.2.2.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.2.2.3

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

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Step 3.1.2.2.3.1

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

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

Step 3.1.2.2.3.2

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

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

Step 3.1.2.2.3.3

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

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

Step 3.1.2.2.3.4

Add $1$ and $1$ .

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

Step 3.1.3

Apply pythagorean identity.

$1 = 1$

Step 4

Since $1 = 1$ , the equation will always be true.

Always true