Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Add $\sec (θ)$ and $\sec (θ)$ .

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

Step 5.2

Add $- \tan (θ)$ and $\tan (θ)$ .

$\frac{2 \sec (θ) + 0}{(\sec (θ) + \tan (θ)) (\sec (θ) - \tan (θ))}$

Step 5.3

Add $2 \sec (θ)$ and $0$ .

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