Trigonometry Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

Add $1$ and $1$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.3

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

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Step 3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

Step 3.1.3.5

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

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

Step 3.1.3.6

Add $1$ and $1$ .

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

Step 3.2

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

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

Step 3.3

Subtract $\sec (x) \tan (x)$ from $- \sec (x) \tan (x)$ .

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