Trigonometry Examples

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

Step 1

Start on the left side.

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

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.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

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

Factor $\cos (x)$ out of $\cos (x) \cos (y)$ .

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

Step 3.3.1.2

Cancel the common factor.

Step 3.3.1.3

Rewrite the expression.

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

Step 3.3.2

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

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Step 3.3.2.1

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

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

Step 3.4

Simplify the denominator.

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Step 3.4.1

Factor $\cos (x) \cos (y)$ out of $\cos (x) \cos (y) \cdot 1 + \cos (x) \cos (y) \frac{\sin (x)}{\cos (x)} \frac{\sin (y)}{\cos (y)}$ .

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Step 3.4.1.1

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

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

Step 3.4.1.2

Factor $\cos (x) \cos (y)$ out of $\cos (x) \cos (y) \frac{\sin (x)}{\cos (x)} \frac{\sin (y)}{\cos (y)}$ .

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

Step 3.4.1.3

Factor $\cos (x) \cos (y)$ out of $\cos (x) \cos (y) (1) + \cos (x) \cos (y) (\frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (y)}{\cos (y)})$ .

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

Step 3.4.2

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{\sin (y)}{\cos (y)}$ .

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

Step 3.4.3

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

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

Step 3.4.4

Combine the numerators over the common denominator.

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

Step 3.4.5

Combine exponents.

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Step 3.4.5.1

Combine $\frac{\cos (x) \cos (y) + \sin (x) \sin (y)}{\cos (x) \cos (y)}$ and $\cos (x)$ .

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

Step 3.4.5.2

Combine $\frac{(\cos (x) \cos (y) + \sin (x) \sin (y)) \cos (x)}{\cos (x) \cos (y)}$ and $\cos (y)$ .

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

Step 3.4.6

Reduce the expression $\frac{(\cos (x) \cos (y) + \sin (x) \sin (y)) \cos (x) \cos (y)}{\cos (x) \cos (y)}$ by cancelling the common factors.

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Step 3.4.6.1

Cancel the common factor.

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

Step 3.4.6.2

Rewrite the expression.

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

Step 3.4.7

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

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Step 3.4.7.1

Cancel the common factor.

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

Step 3.4.7.2

Divide $\cos (x) \cos (y) + \sin (x) \sin (y)$ by $1$ .

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

Step 3.5

Reorder factors in $\frac{\cos (y) \sin (x) + \cos (x) (- \sin (y))}{\cos (x) \cos (y) + \sin (x) \sin (y)}$ .

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Move the negative in front of the fraction.

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

Step 5

Rewrite $- \frac{\cos (x) \sin (y) - \cos (y) \sin (x)}{\cos (x) \cos (y) + \sin (x) \sin (y)}$ as $\frac{\sin (x) \cos (y) - \cos (x) \sin (y)}{\cos (x) \cos (y) + \sin (x) \sin (y)}$ .

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

Step 6

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

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