Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Apply the difference of angles identity.

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Move the negative in front of the fraction.

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

Step 4

Now consider the left side of the equation.

$\tan (x) - \tan (y)$

Step 5

Convert to sines and cosines.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Subtract fractions.

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Step 6.1

To write $\frac{\sin (x)}{\cos (x)}$ as a fraction with a common denominator, multiply by $\frac{\cos (y)}{\cos (y)}$ .

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

Step 6.2

To write $- \frac{\sin (y)}{\cos (y)}$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

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

Step 6.3

Write each expression with a common denominator of $\cos (x) \cos (y)$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.3.1

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

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

Step 6.3.2

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

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

Step 6.3.3

Reorder the factors of $\cos (y) \cos (x)$ .

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

Step 6.4

Combine the numerators over the common denominator.

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Move the negative in front of the fraction.

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

Step 8

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

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