Trigonometry Examples

$\frac{\tan (x)}{1 + \sec (x)} + \frac{1 + \sec (x)}{\tan (x)}$

Step 1

To write $\frac{\tan (x)}{1 + \sec (x)}$ as a fraction with a common denominator, multiply by $\frac{\tan (x)}{\tan (x)}$ .

$\frac{\tan (x)}{1 + \sec (x)} \cdot \frac{\tan (x)}{\tan (x)} + \frac{1 + \sec (x)}{\tan (x)}$

Step 2

To write $\frac{1 + \sec (x)}{\tan (x)}$ as a fraction with a common denominator, multiply by $\frac{1 + \sec (x)}{1 + \sec (x)}$ .

$\frac{\tan (x)}{1 + \sec (x)} \cdot \frac{\tan (x)}{\tan (x)} + \frac{1 + \sec (x)}{\tan (x)} \cdot \frac{1 + \sec (x)}{1 + \sec (x)}$

Step 3

Write each expression with a common denominator of $(1 + \sec (x)) \tan (x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{\tan (x)}{1 + \sec (x)}$ by $\frac{\tan (x)}{\tan (x)}$ .

$\frac{\tan (x) \tan (x)}{(1 + \sec (x)) \tan (x)} + \frac{1 + \sec (x)}{\tan (x)} \cdot \frac{1 + \sec (x)}{1 + \sec (x)}$

Step 3.2

Multiply $\frac{1 + \sec (x)}{\tan (x)}$ by $\frac{1 + \sec (x)}{1 + \sec (x)}$ .

$\frac{\tan (x) \tan (x)}{(1 + \sec (x)) \tan (x)} + \frac{(1 + \sec (x)) (1 + \sec (x))}{\tan (x) (1 + \sec (x))}$

Step 3.3

Reorder the factors of $(1 + \sec (x)) \tan (x)$ .

$\frac{\tan (x) \tan (x)}{\tan (x) (1 + \sec (x))} + \frac{(1 + \sec (x)) (1 + \sec (x))}{\tan (x) (1 + \sec (x))}$

Step 4

Combine the numerators over the common denominator.

$\frac{\tan (x) \tan (x) + (1 + \sec (x)) (1 + \sec (x))}{\tan (x) (1 + \sec (x))}$

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Add $1$ and $1$ .

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

Step 5.2

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

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Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $1$ by $1$ .

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

Step 5.3.1.2

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

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

Step 5.3.1.3

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

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

Step 5.3.1.4

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

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Step 5.3.1.4.1

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

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

Step 5.3.1.4.2

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

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

Step 5.3.1.4.3

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

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

Step 5.3.1.4.4

Add $1$ and $1$ .

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

Step 5.3.2

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

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

Step 5.4

Apply pythagorean identity.

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

Step 5.5

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

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

Factor $2 \sec (x)$ out of $2 \sec (x)$ .

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

Step 5.6.3

Factor $2 \sec (x)$ out of $2 \sec (x) (\sec (x)) + 2 \sec (x) (1)$ .

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

Step 6

Cancel the common factor of $\sec (x) + 1$ and $1 + \sec (x)$ .

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Step 6.1

Reorder terms.

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

$\frac{2 \sec (x)}{\tan (x)}$

Step 7

Separate fractions.

$\frac{2}{1} \cdot \frac{\sec (x)}{\tan (x)}$

Step 8

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{2}{1} \cdot \frac{\sec (x)}{\frac{\sin (x)}{\cos (x)}}$

Step 9

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{2}{1} \cdot \frac{\frac{1}{\cos (x)}}{\frac{\sin (x)}{\cos (x)}}$

Step 10

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

$\frac{2}{1} (\frac{1}{\cos (x)} \cdot \frac{\cos (x)}{\sin (x)})$

Step 11

Cancel the common factor of $\cos (x)$ .

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Step 11.1

Cancel the common factor.

$\frac{2}{1} (\frac{1}{\cos (x)} \cdot \frac{\cos (x)}{\sin (x)})$

Step 11.2

Rewrite the expression.

$\frac{2}{1} \cdot \frac{1}{\sin (x)}$

Step 12

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{2}{1} \csc (x)$

Step 13

Divide $2$ by $1$ .

$2 \csc (x)$