Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

Add $- 1$ and $1$ .

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

Step 5.3

Add $0$ and $2 \sin (x)$ .

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