Trigonometry Examples

$\frac{1}{\csc (θ) - 1} - \frac{1}{\csc (θ) + 1} = 2 \tan^{2} (θ)$

Step 1

Start on the left side.

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

Step 2

Subtract fractions.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 3

Simplify numerator.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

Multiply $- 1$ by $- 1$ .

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

Step 3.2

Subtract $\csc (θ)$ from $\csc (θ)$ .

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

Step 3.3

Add $0$ and $1$ .

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

Step 3.4

Add $1$ and $1$ .

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

Step 4

Simplify denominator.

Tap for more steps...

Step 4.1

Expand $(\csc (θ) + 1) (\csc (θ) - 1)$ using the FOIL Method.

Tap for more steps...

Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

$\frac{2}{\csc^{2} (θ) - 1}$

Step 5

Apply pythagorean identity.

$\frac{2}{\cot^{2} (θ)}$

Step 6

Convert to sines and cosines.

Tap for more steps...

Step 6.1

Write $\cot (θ)$ in sines and cosines using the quotient identity.

$\frac{2}{{(\frac{\cos (θ)}{\sin (θ)})}^{2}}$

Step 6.2

Apply the product rule to $\frac{\cos (θ)}{\sin (θ)}$ .

$\frac{2}{\frac{\cos^{2} (θ)}{\sin^{2} (θ)}}$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Multiply the numerator by the reciprocal of the denominator.

$2 \frac{{\sin (θ)}^{2}}{{\cos (θ)}^{2}}$

Step 7.2

Combine $2$ and $\frac{{\sin (θ)}^{2}}{{\cos (θ)}^{2}}$ .

$\frac{2 \sin^{2} (θ)}{\cos^{2} (θ)}$

Step 8

Rewrite $\frac{2 \sin^{2} (θ)}{\cos^{2} (θ)}$ as $2 \tan^{2} (θ)$ .

$2 \tan^{2} (θ)$

Step 9

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{1}{\csc (θ) - 1} - \frac{1}{\csc (θ) + 1} = 2 \tan^{2} (θ)$ is an identity