Trigonometry Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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Step 1.1.1.1

Apply the distributive property.

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

Step 1.1.1.2

Apply the distributive property.

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

Step 1.1.1.3

Apply the distributive property.

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

Step 1.1.2

Simplify and combine like terms.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

Multiply $1$ by $1$ .

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

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

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

Step 1.1.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 1.1.2.1.5

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

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Step 1.1.2.1.5.1

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

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

Step 1.1.2.1.5.2

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

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

Step 1.1.2.1.5.3

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

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

Step 1.1.2.1.5.4

Add $1$ and $1$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Add $1$ and $0$ .

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

Step 1.2

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

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

Step 2

Apply pythagorean identity.

$1 + 1$

Step 3

Add $1$ and $1$ .

$2$