Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator and denominator of the fraction by $\cos (x)$ .

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Step 3.1.1

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify by cancelling.

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Step 3.3.1

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

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

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

Step 3.3.2

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

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Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

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

Step 3.4

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

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

Step 3.5

Simplify the denominator.

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

Step 4

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

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