Precalculus Examples

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

Step 1

Start on the left side.

$\frac{\csc (x)}{\sec (x) - \tan (x)} - \frac{\csc (x)}{\sec (x) + \tan (x)}$

Step 2

Subtract fractions.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

$\frac{\csc (x) (\sec (x) + \tan (x))}{(\sec (x) - \tan (x)) (\sec (x) + \tan (x))} - \frac{\csc (x) (\sec (x) - \tan (x))}{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}$

Step 2.3.3

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

$\frac{\csc (x) (\sec (x) + \tan (x))}{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))} - \frac{\csc (x) (\sec (x) - \tan (x))}{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{\csc (x) (\sec (x) + \tan (x)) - \csc (x) (\sec (x) - \tan (x))}{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}$

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.

$\frac{\csc (x) \sec (x) + \csc (x) \tan (x) - \csc (x) (\sec (x) - \tan (x))}{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}$

Step 3.1.2

Apply the distributive property.

$\frac{\csc (x) \sec (x) + \csc (x) \tan (x) - \csc (x) \sec (x) - \csc (x) (- \tan (x))}{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}$

Step 3.1.3

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

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Step 3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.3.2

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

$\frac{\csc (x) \sec (x) + \csc (x) \tan (x) - \csc (x) \sec (x) + \csc (x) \tan (x)}{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}$

Step 3.2

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

$\frac{\csc (x) \tan (x) + 0 + \csc (x) \tan (x)}{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}$

Step 3.3

Add $\csc (x) \tan (x)$ and $0$ .

$\frac{\csc (x) \tan (x) + \csc (x) \tan (x)}{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}$

Step 3.4

Add $\csc (x) \tan (x)$ and $\csc (x) \tan (x)$ .

$\frac{2 \csc (x) \tan (x)}{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}$

Step 4

Simplify denominator.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

$\frac{2 \csc (x) \tan (x)}{\sec (x) (\sec (x) - \tan (x)) + \tan (x) (\sec (x) - \tan (x))}$

Step 4.1.2

Apply the distributive property.

$\frac{2 \csc (x) \tan (x)}{\sec (x) \sec (x) + \sec (x) (- \tan (x)) + \tan (x) (\sec (x) - \tan (x))}$

Step 4.1.3

Apply the distributive property.

$\frac{2 \csc (x) \tan (x)}{\sec (x) \sec (x) + \sec (x) (- \tan (x)) + \tan (x) \sec (x) + \tan (x) (- \tan (x))}$

Step 4.2

Simplify and combine like terms.

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

Step 5

Apply pythagorean identity.

$\frac{2 \csc (x) \tan (x)}{1}$

Step 6

Convert to sines and cosines.

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Step 6.1

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

$\frac{2 \frac{1}{\sin (x)} \tan (x)}{1}$

Step 6.2

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

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

Step 7

Simplify.

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Step 7.1

Divide $2 \frac{1}{\sin (x)} \frac{\sin (x)}{\cos (x)}$ by $1$ .

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

Step 7.2

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

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

Step 7.3

Cancel the common factor of $\sin (x)$ .

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Step 7.3.1

Cancel the common factor.

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

Step 7.3.2

Rewrite the expression.

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

Step 7.4

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

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

Step 8

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

$2 \sec (x)$

Step 9

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

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