Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify by cancelling.

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Step 3.3.1

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

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

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

Step 3.3.2

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

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Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.5

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

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

Step 4

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

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