Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $- - \tan (x)$ .

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Step 5.2.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.2

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

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

Step 5.3

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

$\frac{0 + \tan (x) + \tan (x)}{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}$

Step 5.4

Add $0$ and $\tan (x)$ .

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

Step 5.5

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

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