Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

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

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

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

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

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Move $- 1$ to the left of $\cos (x)$ .

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

Step 3.4.2

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

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

Step 3.5

Factor $\cos (x)$ out of $\cos (x) \cdot 1 + \cos (x) (- \cos (x))$ .

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Step 3.5.1

Factor $\cos (x)$ out of $\cos (x) \cdot 1$ .

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

Step 3.5.2

Factor $\cos (x)$ out of $\cos (x) (1) + \cos (x) (- \cos (x))$ .

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

Step 3.6

Cancel the common factor of $1 - \cos (x)$ .

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

Step 4

Rewrite $\frac{1}{\cos (x)}$ as $\sec (x)$ .

$\sec (x)$

Step 5

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

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