Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(1 + \tan (x))}^{2}$ as $(1 + \tan (x)) (1 + \tan (x))$ .

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $1$ by $1$ .

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

Step 1.3.1.2

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

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

Step 1.3.1.3

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

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

Step 1.3.1.4

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

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Step 1.3.1.4.1

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

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

Step 1.3.1.4.2

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

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

Step 1.3.1.4.3

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

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

Step 1.3.1.4.4

Add $1$ and $1$ .

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

Step 1.3.2

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

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

Step 1.4

Rewrite ${(1 - \tan (x))}^{2}$ as $(1 - \tan (x)) (1 - \tan (x))$ .

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

Step 1.5

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

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

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

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $1$ by $1$ .

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

Step 1.6.1.2

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

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

Step 1.6.1.3

Multiply $- 1$ by $1$ .

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

Step 1.6.1.4

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

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Step 1.6.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.6.1.4.2

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

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

Step 1.6.1.4.3

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

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

Step 1.6.1.4.4

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

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

Step 1.6.1.4.5

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

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

Step 1.6.1.4.6

Add $1$ and $1$ .

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

Step 1.6.2

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

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $1 + 2 \tan (x) + \tan^{2} (x) + 1 - 2 \tan (x) + \tan^{2} (x)$ .

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Step 2.1.1

Subtract $2 \tan (x)$ from $2 \tan (x)$ .

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

Step 2.1.2

Add $1$ and $0$ .

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

Step 2.2

Add $1$ and $1$ .

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

Step 2.3

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

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

Step 2.4

Factor $2$ out of $2$ .

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

Step 2.5

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

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

Step 2.6

Factor $2$ out of $2 (1) + 2 (\tan^{2} (x))$ .

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

Step 3

Rearrange terms.

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

Step 4

Apply pythagorean identity.

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