Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(\frac{1}{\cos (x)} - 1) (\frac{1}{\cos (x)} + 1)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Factor $\cos (x)$ out of $\cos (x) (\frac{1}{\cos (x)} - 1) + \cos (x) (\frac{1}{\cos (x)} + 1)$ .

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

Step 6.2

Add $\frac{1}{\cos (x)}$ and $\frac{1}{\cos (x)}$ .

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

Step 6.3

Add $- 1$ and $1$ .

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

Step 6.4

Add $2 \frac{1}{\cos (x)}$ and $0$ .

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

Step 6.5

Combine exponents.

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Step 6.5.1

Combine $\cos (x)$ and $\frac{1}{\cos (x)}$ .

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

Step 6.5.2

Combine $2$ and $\frac{\cos (x)}{\cos (x)}$ .

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

Step 6.6

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

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Step 6.6.1

Cancel the common factor.

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

Step 6.6.2

Divide $2$ by $1$ .

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

Step 7

Simplify the denominator.

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Step 7.1

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

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

Step 7.2

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

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