Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Subtract fractions.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 3

Simplify numerator.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

Multiply $\csc (x)$ by $1$ .

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

Step 3.1.3

Multiply $\csc (x) (- \csc (x))$ .

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Step 3.1.3.1

Raise $\csc (x)$ to the power of $1$ .

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

Step 3.1.3.2

Raise $\csc (x)$ to the power of $1$ .

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

Step 3.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.1.3.4

Add $1$ and $1$ .

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

Step 3.1.4

Apply the distributive property.

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

Step 3.1.5

Multiply $- 1$ by $1$ .

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

Step 3.1.6

Multiply $- \csc (x) \csc (x)$ .

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Step 3.1.6.1

Raise $\csc (x)$ to the power of $1$ .

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

Step 3.1.6.2

Raise $\csc (x)$ to the power of $1$ .

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

Step 3.1.6.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.1.6.4

Add $1$ and $1$ .

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

Step 3.2

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

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

Step 3.3

Add $- {\csc (x)}^{2}$ and $0$ .

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

Step 3.4

Subtract ${\csc (x)}^{2}$ from $- {\csc (x)}^{2}$ .

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

Step 4

Simplify denominator.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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

Step 5

Apply Pythagorean identity.

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Step 5.1

Reorder $1$ and $- \csc^{2} (x)$ .

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

Step 5.2

Factor $- 1$ out of $- \csc^{2} (x)$ .

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

Step 5.3

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 5.4

Factor $- 1$ out of $- \csc^{2} (x) - 1 \cdot - 1$ .

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

Step 5.5

Apply pythagorean identity.

$\frac{- 2 \csc^{2} (x)}{- \cot^{2} (x)}$

Step 6

Convert to sines and cosines.

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Step 6.1

Apply the reciprocal identity to $\csc (x)$ .

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

Step 6.2

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

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

Step 6.3

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 6.4

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

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

Step 7

Simplify.

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Step 7.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.2

One to any power is one.

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

Step 7.3

Combine $- 2$ and $\frac{1}{{\sin (x)}^{2}}$ .

$\frac{- 2}{{\sin (x)}^{2}} (- \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}})$

Step 7.4

Cancel the common factor of ${\sin (x)}^{2}$ .

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Step 7.4.1

Move the leading negative in $- \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}}$ into the numerator.

$\frac{- 2}{{\sin (x)}^{2}} \cdot \frac{- {\sin (x)}^{2}}{{\cos (x)}^{2}}$

Step 7.4.2

Factor ${\sin (x)}^{2}$ out of $- {\sin (x)}^{2}$ .

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

Step 7.4.3

Cancel the common factor.

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

Step 7.4.4

Rewrite the expression.

$- 2 \frac{- 1}{{\cos (x)}^{2}}$

Step 7.5

Combine $- 2$ and $\frac{- 1}{{\cos (x)}^{2}}$ .

$\frac{- 2 \cdot - 1}{{\cos (x)}^{2}}$

Step 7.6

Multiply $- 2$ by $- 1$ .

$\frac{2}{\cos^{2} (x)}$

Step 8

Rewrite $\frac{2}{\cos^{2} (x)}$ as $2 \sec^{2} (x)$ .

$2 \sec^{2} (x)$

Step 9

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

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