Verify the Identity 1/(1+cos(x))-1/(1-cos(x))=-2csc(x)cot(x)

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

Start on the left side.

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

Subtract fractions.

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

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

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

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

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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Combine.

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

Combine.

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

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

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

Combine the numerators over the common denominator.

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

Simplify numerator.

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Simplify each term.

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Multiply $1 - \cos (x)$ by $1$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

Subtract $1$ from $1$ .

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

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

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

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

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

Simplify denominator.

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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

Apply pythagorean identity.

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

Move the negative in front of the fraction.

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

Now consider the right side of the equation.

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

Convert to sines and cosines.

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

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

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

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

Simplify.

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

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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