Trigonometry Examples

$\frac{5 \cos (x)}{1 + \sin (x)} + \frac{5 + 5 \sin (x)}{\cos (x)}$

Step 1

Factor $5$ out of $5 + 5 \sin (x)$ .

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Step 1.1

Factor $5$ out of $5$ .

$\frac{5 \cos (x)}{1 + \sin (x)} + \frac{5 \cdot 1 + 5 \sin (x)}{\cos (x)}$

Step 1.2

Factor $5$ out of $5 \cdot 1 + 5 \sin (x)$ .

$\frac{5 \cos (x)}{1 + \sin (x)} + \frac{5 (1 + \sin (x))}{\cos (x)}$

Step 2

To write $\frac{5 \cos (x)}{1 + \sin (x)}$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

$\frac{5 \cos (x)}{1 + \sin (x)} \cdot \frac{\cos (x)}{\cos (x)} + \frac{5 (1 + \sin (x))}{\cos (x)}$

Step 3

To write $\frac{5 (1 + \sin (x))}{\cos (x)}$ as a fraction with a common denominator, multiply by $\frac{1 + \sin (x)}{1 + \sin (x)}$ .

$\frac{5 \cos (x)}{1 + \sin (x)} \cdot \frac{\cos (x)}{\cos (x)} + \frac{5 (1 + \sin (x))}{\cos (x)} \cdot \frac{1 + \sin (x)}{1 + \sin (x)}$

Step 4

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

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Step 4.1

Multiply $\frac{5 \cos (x)}{1 + \sin (x)}$ by $\frac{\cos (x)}{\cos (x)}$ .

$\frac{5 \cos (x) \cos (x)}{(1 + \sin (x)) \cos (x)} + \frac{5 (1 + \sin (x))}{\cos (x)} \cdot \frac{1 + \sin (x)}{1 + \sin (x)}$

Step 4.2

Multiply $\frac{5 (1 + \sin (x))}{\cos (x)}$ by $\frac{1 + \sin (x)}{1 + \sin (x)}$ .

$\frac{5 \cos (x) \cos (x)}{(1 + \sin (x)) \cos (x)} + \frac{5 (1 + \sin (x)) (1 + \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 4.3

Reorder the factors of $(1 + \sin (x)) \cos (x)$ .

$\frac{5 \cos (x) \cos (x)}{\cos (x) (1 + \sin (x))} + \frac{5 (1 + \sin (x)) (1 + \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 5

Combine the numerators over the common denominator.

$\frac{5 \cos (x) \cos (x) + 5 (1 + \sin (x)) (1 + \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $5$ out of $5 \cos (x) \cos (x) + 5 (1 + \sin (x)) (1 + \sin (x))$ .

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Step 6.1.1

Factor $5$ out of $5 \cos (x) \cos (x)$ .

$\frac{5 (\cos (x) \cos (x)) + 5 (1 + \sin (x)) (1 + \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 6.1.2

Factor $5$ out of $5 (1 + \sin (x)) (1 + \sin (x))$ .

$\frac{5 (\cos (x) \cos (x)) + 5 ((1 + \sin (x)) (1 + \sin (x)))}{\cos (x) (1 + \sin (x))}$

Step 6.1.3

Factor $5$ out of $5 (\cos (x) \cos (x)) + 5 ((1 + \sin (x)) (1 + \sin (x)))$ .

$\frac{5 (\cos (x) \cos (x) + (1 + \sin (x)) (1 + \sin (x)))}{\cos (x) (1 + \sin (x))}$

Step 6.2

Multiply $\cos (x) \cos (x)$ .

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Step 6.2.1

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

$\frac{5 (\cos^{1} (x) \cos (x) + (1 + \sin (x)) (1 + \sin (x)))}{\cos (x) (1 + \sin (x))}$

Step 6.2.2

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

$\frac{5 (\cos^{1} (x) \cos^{1} (x) + (1 + \sin (x)) (1 + \sin (x)))}{\cos (x) (1 + \sin (x))}$

Step 6.2.3

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

$\frac{5 ({\cos (x)}^{1 + 1} + (1 + \sin (x)) (1 + \sin (x)))}{\cos (x) (1 + \sin (x))}$

Step 6.2.4

Add $1$ and $1$ .

$\frac{5 (\cos^{2} (x) + (1 + \sin (x)) (1 + \sin (x)))}{\cos (x) (1 + \sin (x))}$

Step 6.3

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

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Step 6.3.1

Apply the distributive property.

$\frac{5 (\cos^{2} (x) + 1 (1 + \sin (x)) + \sin (x) (1 + \sin (x)))}{\cos (x) (1 + \sin (x))}$

Step 6.3.2

Apply the distributive property.

$\frac{5 (\cos^{2} (x) + 1 \cdot 1 + 1 \sin (x) + \sin (x) (1 + \sin (x)))}{\cos (x) (1 + \sin (x))}$

Step 6.3.3

Apply the distributive property.

$\frac{5 (\cos^{2} (x) + 1 \cdot 1 + 1 \sin (x) + \sin (x) \cdot 1 + \sin (x) \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 6.4

Simplify and combine like terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

Multiply $1$ by $1$ .

$\frac{5 (\cos^{2} (x) + 1 + 1 \sin (x) + \sin (x) \cdot 1 + \sin (x) \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 6.4.1.2

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

$\frac{5 (\cos^{2} (x) + 1 + \sin (x) + \sin (x) \cdot 1 + \sin (x) \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 6.4.1.3

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

$\frac{5 (\cos^{2} (x) + 1 + \sin (x) + \sin (x) + \sin (x) \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 6.4.1.4

Multiply $\sin (x) \sin (x)$ .

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Step 6.4.1.4.1

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

$\frac{5 (\cos^{2} (x) + 1 + \sin (x) + \sin (x) + \sin^{1} (x) \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 6.4.1.4.2

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

$\frac{5 (\cos^{2} (x) + 1 + \sin (x) + \sin (x) + \sin^{1} (x) \sin^{1} (x))}{\cos (x) (1 + \sin (x))}$

Step 6.4.1.4.3

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

$\frac{5 (\cos^{2} (x) + 1 + \sin (x) + \sin (x) + {\sin (x)}^{1 + 1})}{\cos (x) (1 + \sin (x))}$

Step 6.4.1.4.4

Add $1$ and $1$ .

$\frac{5 (\cos^{2} (x) + 1 + \sin (x) + \sin (x) + \sin^{2} (x))}{\cos (x) (1 + \sin (x))}$

Step 6.4.2

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

$\frac{5 (\cos^{2} (x) + 1 + 2 \sin (x) + \sin^{2} (x))}{\cos (x) (1 + \sin (x))}$

Step 6.5

Rewrite $\cos^{2} (x) + 1 + 2 \sin (x) + \sin^{2} (x)$ in a factored form.

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Step 6.5.1

Regroup terms.

$\frac{5 (\cos^{2} (x) + \sin^{2} (x) + 1 + 2 \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 6.5.2

Rearrange terms.

$\frac{5 (\sin^{2} (x) + \cos^{2} (x) + 1 + 2 \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 6.5.3

Apply pythagorean identity.

$\frac{5 (1 + 1 + 2 \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 6.5.4

Add $1$ and $1$ .

$\frac{5 (2 + 2 \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 6.5.5

Factor $2$ out of $2 + 2 \sin (x)$ .

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Step 6.5.5.1

Factor $2$ out of $2$ .

$\frac{5 (2 \cdot 1 + 2 \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 6.5.5.2

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

$\frac{5 (2 (1 + \sin (x)))}{\cos (x) (1 + \sin (x))}$

$\frac{5 \cdot 2 (1 + \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 6.6

Multiply $5$ by $2$ .

$\frac{10 (1 + \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 7

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

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Step 7.1

Cancel the common factor.

$\frac{10 (1 + \sin (x))}{\cos (x) (1 + \sin (x))}$

Step 7.2

Rewrite the expression.

$\frac{10}{\cos (x)}$

Step 8

Separate fractions.

$\frac{10}{1} \cdot \frac{1}{\cos (x)}$

Step 9

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{10}{1} \sec (x)$

Step 10

Divide $10$ by $1$ .

$10 \sec (x)$