Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Add $1$ and $1$ .

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

Step 5.2

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

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Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Apply the distributive property.

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

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 $1$ .

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

Step 5.3.1.2

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

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

Step 5.3.1.3

Multiply $- 1$ by $1$ .

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

Step 5.3.1.4

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

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Step 5.3.1.4.1

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

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

Step 5.3.1.4.2

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

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

Step 5.3.1.4.3

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

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

Step 5.3.1.4.4

Add $1$ and $1$ .

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

Step 5.3.2

Subtract $\sin (x)$ from $\sin (x)$ .

$\frac{\cos^{2} (x) + 1 + 0 - \sin^{2} (x)}{\cos (x) (1 + \sin (x))}$

Step 5.3.3

Add $1$ and $0$ .

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

Step 5.4

Apply pythagorean identity.

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

Step 5.5

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

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

Step 6

Cancel the common factor of $\cos^{2} (x)$ and $\cos (x)$ .

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Step 6.1

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

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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