Precalculus Examples

\frac{cos (a + b)}{cos (a) sin (b)} = cot (b) - tan (a)

$\frac{\cos (a + b)}{\cos (a) \sin (b)} = \cot (b) - \tan (a)$

Step 1

Start on the left side.

\frac{cos (a + b)}{cos (a) sin (b)}

$\frac{\cos (a + b)}{\cos (a) \sin (b)}$

Step 2

Apply the sum of angles identity

cos (x + y) = cos (x) cos (y) - sin (x) sin (y)

$\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

\frac{cos (a) cos (b) - sin (a) sin (b)}{cos (a) sin (b)}

$\frac{\cos (a) \cos (b) - \sin (a) \sin (b)}{\cos (a) \sin (b)}$

Step 3

Simplify.

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Step 3.1

Factor

- 1

$- 1$ out of

- sin (a) sin (b)

$- \sin (a) \sin (b)$ .

\frac{- (sin (a) sin (b)) + cos (a) cos (b)}{cos (a) sin (b)}

$\frac{- (\sin (a) \sin (b)) + \cos (a) \cos (b)}{\cos (a) \sin (b)}$

Step 3.2

Factor

- 1

$- 1$ out of

cos (a) cos (b)

$\cos (a) \cos (b)$ .

\frac{- (sin (a) sin (b)) - (- cos (a) cos (b))}{cos (a) sin (b)}

$\frac{- (\sin (a) \sin (b)) - (- \cos (a) \cos (b))}{\cos (a) \sin (b)}$

Step 3.3

Factor

- 1

$- 1$ out of

- (sin (a) sin (b)) - (- cos (a) cos (b))

$- (\sin (a) \sin (b)) - (- \cos (a) \cos (b))$ .

\frac{- (sin (a) sin (b) - cos (a) cos (b))}{cos (a) sin (b)}

$\frac{- (\sin (a) \sin (b) - \cos (a) \cos (b))}{\cos (a) \sin (b)}$

Step 3.4

Rewrite

- (sin (a) sin (b) - cos (a) cos (b))

$- (\sin (a) \sin (b) - \cos (a) \cos (b))$ as

- 1 (sin (a) sin (b) - cos (a) cos (b))

$- 1 (\sin (a) \sin (b) - \cos (a) \cos (b))$ .

\frac{- 1 (sin (a) sin (b) - cos (a) cos (b))}{cos (a) sin (b)}

$\frac{- 1 (\sin (a) \sin (b) - \cos (a) \cos (b))}{\cos (a) \sin (b)}$

Step 3.5

Move the negative in front of the fraction.

- \frac{sin (a) sin (b) - cos (a) cos (b)}{cos (a) sin (b)}

$- \frac{\sin (a) \sin (b) - \cos (a) \cos (b)}{\cos (a) \sin (b)}$

- \frac{sin (a) sin (b) - cos (a) cos (b)}{cos (a) sin (b)}

$- \frac{\sin (a) \sin (b) - \cos (a) \cos (b)}{\cos (a) \sin (b)}$

Step 4

Now consider the right side of the equation.

cot (b) - tan (a)

$\cot (b) - \tan (a)$

Step 5

Convert to sines and cosines.

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Step 5.1

Write

cot (b)

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

\frac{cos (b)}{sin (b)} - tan (a)

$\frac{\cos (b)}{\sin (b)} - \tan (a)$

Step 5.2

Write

tan (a)

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

\frac{cos (b)}{sin (b)} - \frac{sin (a)}{cos (a)}

$\frac{\cos (b)}{\sin (b)} - \frac{\sin (a)}{\cos (a)}$

\frac{cos (b)}{sin (b)} - \frac{sin (a)}{cos (a)}

$\frac{\cos (b)}{\sin (b)} - \frac{\sin (a)}{\cos (a)}$

Step 6

Subtract fractions.

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Step 6.1

To write

\frac{cos (b)}{sin (b)}

$\frac{\cos (b)}{\sin (b)}$ as a fraction with a common denominator, multiply by

\frac{cos (a)}{cos (a)}

$\frac{\cos (a)}{\cos (a)}$ .

\frac{cos (b)}{sin (b)} \cdot \frac{cos (a)}{cos (a)} - \frac{sin (a)}{cos (a)}

$\frac{\cos (b)}{\sin (b)} \cdot \frac{\cos (a)}{\cos (a)} - \frac{\sin (a)}{\cos (a)}$

Step 6.2

To write

- \frac{sin (a)}{cos (a)}

$- \frac{\sin (a)}{\cos (a)}$ as a fraction with a common denominator, multiply by

\frac{sin (b)}{sin (b)}

$\frac{\sin (b)}{\sin (b)}$ .

$\frac{\cos (b)}{\sin (b)} \cdot \frac{\cos (a)}{\cos (a)} - \frac{\sin (a)}{\cos (a)} \cdot \frac{\sin (b)}{\sin (b)}$

Step 6.3

Write each expression with a common denominator of

$\sin (b) \cos (a)$ , by multiplying each by an appropriate factor of

$1$ .

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Step 6.3.1

Multiply

$\frac{\cos (b)}{\sin (b)}$ by

$\frac{\cos (a)}{\cos (a)}$ .

$\frac{\cos (b) \cos (a)}{\sin (b) \cos (a)} - \frac{\sin (a)}{\cos (a)} \cdot \frac{\sin (b)}{\sin (b)}$

Step 6.3.2

Multiply

$\frac{\sin (a)}{\cos (a)}$ by

$\frac{\sin (b)}{\sin (b)}$ .

$\frac{\cos (b) \cos (a)}{\sin (b) \cos (a)} - \frac{\sin (a) \sin (b)}{\cos (a) \sin (b)}$

Step 6.3.3

Reorder the factors of

$\sin (b) \cos (a)$ .

$\frac{\cos (b) \cos (a)}{\cos (a) \sin (b)} - \frac{\sin (a) \sin (b)}{\cos (a) \sin (b)}$

Step 6.4

Combine the numerators over the common denominator.

$\frac{\cos (b) \cos (a) - \sin (a) \sin (b)}{\cos (a) \sin (b)}$

Step 7

Simplify.

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Step 7.1

Factor

$- 1$ out of

$- \sin (a) \sin (b)$ .

$\frac{- (\sin (a) \sin (b)) + \cos (a) \cos (b)}{\cos (a) \sin (b)}$

Step 7.2

Factor

$- 1$ out of

$\cos (a) \cos (b)$ .

$\frac{- (\sin (a) \sin (b)) - (- \cos (a) \cos (b))}{\cos (a) \sin (b)}$

Step 7.3

Factor

$- 1$ out of

$- (\sin (a) \sin (b)) - (- \cos (a) \cos (b))$ .

$\frac{- (\sin (a) \sin (b) - \cos (a) \cos (b))}{\cos (a) \sin (b)}$

Step 7.4

Rewrite

$- (\sin (a) \sin (b) - \cos (a) \cos (b))$ as

$- 1 (\sin (a) \sin (b) - \cos (a) \cos (b))$ .

$\frac{- 1 (\sin (a) \sin (b) - \cos (a) \cos (b))}{\cos (a) \sin (b)}$

Step 7.5

Move the negative in front of the fraction.

$- \frac{\sin (a) \sin (b) - \cos (a) \cos (b)}{\cos (a) \sin (b)}$

Step 8

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

$\frac{\cos (a + b)}{\cos (a) \sin (b)} = \cot (b) - \tan (a)$ is an identity