Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Subtract fractions.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

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.

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

Step 3.1.2

Multiply $- 1$ by $1$ .

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

Step 3.2

Subtract $1$ from $1$ .

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

Step 3.3

Subtract $\cos (x)$ from $0$ .

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

Step 3.4

Subtract $\cos (x)$ from $- \cos (x)$ .

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

Step 4

Simplify denominator.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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

Step 5

Apply pythagorean identity.

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

Step 6

Move the negative in front of the fraction.

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

Step 7

Write $- 1$ as a fraction with denominator $1$ .

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

Step 8

Combine.

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

Step 9

Multiply $2$ by $- 1$ .

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

Step 10

Multiply ${\sin (x)}^{2}$ by $1$ .

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

Step 11

Move the negative in front of the fraction.

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

Step 12

Now consider the right side of the equation.

$- 2 \csc (x) \cot (x)$

Step 13

Convert to sines and cosines.

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Step 13.1

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

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

Step 13.2

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

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

Step 14

Simplify.

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Step 14.1

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

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

Step 14.2

Move the negative in front of the fraction.

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

Step 14.3

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

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Step 14.3.1

Multiply $\frac{\cos (x)}{\sin (x)}$ by $\frac{2}{\sin (x)}$ .

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

Step 14.3.2

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

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

Step 14.3.3

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

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

Step 14.3.4

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

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

Step 14.3.5

Add $1$ and $1$ .

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

Step 14.4

Move $2$ to the left of $\cos (x)$ .

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

Step 15

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

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