Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

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

Factor $\cos (A)$ out of $\cos (A) \sin (A)$ .

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

Step 3.3.1.2

Cancel the common factor.

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

Step 3.3.1.3

Rewrite the expression.

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

Step 3.3.2

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

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Step 3.3.2.1

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

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

Step 3.3.3

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

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Step 3.3.3.1

Factor $\cos (A)$ out of $\cos (A) \sin (A)$ .

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

Step 3.3.3.2

Cancel the common factor.

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

Step 3.3.3.3

Rewrite the expression.

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

Step 3.3.4

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

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Step 3.3.4.1

Cancel the common factor.

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

Step 3.3.4.2

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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

Step 4

Reorder terms.

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

Step 5

Rewrite $\frac{\sin (A) - \cos (A)}{\cos (A) + \sin (A)}$ as $\frac{\tan (A) - 1}{\tan (A) + 1}$ .

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

Step 6

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

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