Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Add fractions.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 3

Simplify numerator.

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Step 3.1

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

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

Step 3.2

Subtract $1$ from $1$ .

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

Step 3.3

Add $2 \sec (x)$ and $0$ .

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

Step 4

Simplify denominator.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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

Step 5

Apply pythagorean identity.

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

Step 6

Convert to sines and cosines.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

Simplify.

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Step 7.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.2

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

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

Step 7.3

Combine.

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

Step 7.4

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

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

Step 8

Rewrite $\frac{2 \cos (x)}{\sin^{2} (x)}$ as $2 \cot (x) \csc (x)$ .

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

Step 9

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

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