Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $- 1$ by $- 1$ .

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

Step 5.3

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

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

Step 5.4

Add $0$ and $1$ .

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

Step 5.5

Add $1$ and $1$ .

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