Trigonometry Examples

$3 (\sec (x) + \tan (x)) (\sec (x) - \tan (x))$

Step 1

Apply the distributive property.

$(3 \sec (x) + 3 \tan (x)) (\sec (x) - \tan (x))$

Step 2

Expand $(3 \sec (x) + 3 \tan (x)) (\sec (x) - \tan (x))$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$3 \sec (x) (\sec (x) - \tan (x)) + 3 \tan (x) (\sec (x) - \tan (x))$

Step 2.2

Apply the distributive property.

$3 \sec (x) \sec (x) + 3 \sec (x) (- \tan (x)) + 3 \tan (x) (\sec (x) - \tan (x))$

Step 2.3

Apply the distributive property.

$3 \sec (x) \sec (x) + 3 \sec (x) (- \tan (x)) + 3 \tan (x) \sec (x) + 3 \tan (x) (- \tan (x))$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $3 \sec (x) \sec (x)$ .

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Step 3.1.1.1

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

$3 (\sec^{1} (x) \sec (x)) + 3 \sec (x) (- \tan (x)) + 3 \tan (x) \sec (x) + 3 \tan (x) (- \tan (x))$

Step 3.1.1.2

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

$3 (\sec^{1} (x) \sec^{1} (x)) + 3 \sec (x) (- \tan (x)) + 3 \tan (x) \sec (x) + 3 \tan (x) (- \tan (x))$

Step 3.1.1.3

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

$3 {\sec (x)}^{1 + 1} + 3 \sec (x) (- \tan (x)) + 3 \tan (x) \sec (x) + 3 \tan (x) (- \tan (x))$

Step 3.1.1.4

Add $1$ and $1$ .

$3 \sec^{2} (x) + 3 \sec (x) (- \tan (x)) + 3 \tan (x) \sec (x) + 3 \tan (x) (- \tan (x))$

Step 3.1.2

Multiply $- 1$ by $3$ .

$3 \sec^{2} (x) - 3 \sec (x) \tan (x) + 3 \tan (x) \sec (x) + 3 \tan (x) (- \tan (x))$

Step 3.1.3

Multiply $3 \tan (x) (- \tan (x))$ .

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Step 3.1.3.1

Multiply $- 1$ by $3$ .

$3 \sec^{2} (x) - 3 \sec (x) \tan (x) + 3 \tan (x) \sec (x) - 3 \tan (x) \tan (x)$

Step 3.1.3.2

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

$3 \sec^{2} (x) - 3 \sec (x) \tan (x) + 3 \tan (x) \sec (x) - 3 (\tan^{1} (x) \tan (x))$

Step 3.1.3.3

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

$3 \sec^{2} (x) - 3 \sec (x) \tan (x) + 3 \tan (x) \sec (x) - 3 (\tan^{1} (x) \tan^{1} (x))$

Step 3.1.3.4

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

$3 \sec^{2} (x) - 3 \sec (x) \tan (x) + 3 \tan (x) \sec (x) - 3 {\tan (x)}^{1 + 1}$

Step 3.1.3.5

Add $1$ and $1$ .

$3 \sec^{2} (x) - 3 \sec (x) \tan (x) + 3 \tan (x) \sec (x) - 3 \tan^{2} (x)$

Step 3.2

Reorder the factors of $3 \tan (x) \sec (x)$ .

$3 \sec^{2} (x) - 3 \sec (x) \tan (x) + 3 \sec (x) \tan (x) - 3 \tan^{2} (x)$

Step 3.3

Add $- 3 \sec (x) \tan (x)$ and $3 \sec (x) \tan (x)$ .

$3 \sec^{2} (x) + 0 - 3 \tan^{2} (x)$

Step 3.4

Add $3 \sec^{2} (x)$ and $0$ .

$3 \sec^{2} (x) - 3 \tan^{2} (x)$

Step 4

Simplify with factoring out.

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Step 4.1

Factor $3$ out of $3 \sec^{2} (x)$ .

$3 (\sec^{2} (x)) - 3 \tan^{2} (x)$

Step 4.2

Factor $3$ out of $- 3 \tan^{2} (x)$ .

$3 (\sec^{2} (x)) + 3 (- \tan^{2} (x))$

Step 4.3

Factor $3$ out of $3 (\sec^{2} (x)) + 3 (- \tan^{2} (x))$ .

$3 (\sec^{2} (x) - \tan^{2} (x))$

Step 5

Apply pythagorean identity.

$3 \cdot 1$

Step 6

Multiply $3$ by $1$ .

$3$