Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify by cancelling.

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Step 3.3.1

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

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Step 3.3.1.1

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

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

Step 3.3.1.2

Cancel the common factor.

Step 3.3.1.3

Rewrite the expression.

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

Step 3.3.2

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

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Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

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

Step 3.4

Simplify the denominator.

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Move the negative in front of the fraction.

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

Step 5

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

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

Step 6

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

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