Verify the Identity (cos(x))/(sec(x)-1)-(cos(x))/(sec(x)+1)=(2cos(x))/(tan(x)^2)

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

Start on the left side.

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

Subtract fractions.

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

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

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

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

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

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Combine.

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

Combine.

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

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

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

Combine the numerators over the common denominator.

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

Simplify numerator.

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Simplify each term.

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Apply the distributive property.

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

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

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

Apply the distributive property.

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

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

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Multiply $- 1$ by $- 1$ .

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

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

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

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

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

Add $0$ and $\cos (x)$ .

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

Add $\cos (x)$ and $\cos (x)$ .

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

Simplify denominator.

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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

Apply pythagorean identity.

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

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

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

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