Precalculus Examples

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

Step 1

To write $\frac{2}{1 - \sin (x)}$ as a fraction with a common denominator, multiply by $\frac{1 + \sin (x)}{1 + \sin (x)}$ .

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

Step 2

To write $\frac{2}{1 + \sin (x)}$ as a fraction with a common denominator, multiply by $\frac{1 - \sin (x)}{1 - \sin (x)}$ .

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

Step 3

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

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Step 3.1

Multiply $\frac{2}{1 - \sin (x)}$ by $\frac{1 + \sin (x)}{1 + \sin (x)}$ .

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

Step 3.2

Multiply $\frac{2}{1 + \sin (x)}$ by $\frac{1 - \sin (x)}{1 - \sin (x)}$ .

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

Add $1$ and $1$ .

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

Step 5.3

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

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

Step 5.4

Add $2$ and $0$ .

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

Step 6

Multiply $2$ by $2$ .

$\frac{4}{(1 + \sin (x)) (1 - \sin (x))}$