Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 2.2

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

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.

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

Step 3.3.1.2

Rewrite the expression.

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

Step 3.3.2

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

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Step 3.3.2.1

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

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

Step 3.4

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

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

Step 3.5

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

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

Step 4

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

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