Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

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

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

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

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

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Step 3.3.1

Move the leading negative in $- \frac{1}{\cos (x)}$ into the numerator.

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

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

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

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 $\cos (x) \cdot 1$ .

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

Step 3.5.2

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

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 3.6.3

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

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

Step 3.6.4

Reorder terms.

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

Step 3.6.5

Cancel the common factor.

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

Step 3.6.6

Rewrite the expression.

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

Step 3.7

Move the negative in front of the fraction.

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

Step 4

Write $- 1$ as a fraction with denominator $1$ .

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

Step 5

Combine.

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

Step 6

Multiply $- 1$ by $1$ .

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

Step 7

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

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

Step 8

Move the negative in front of the fraction.

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

Step 9

Now consider the right side of the equation.

$- \sec (x)$

Step 10

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

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

Step 11

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

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