Precalculus Examples

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

Step 1

Start on the left side.

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

Step 2

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

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

Step 3

Simplify.

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Step 3.1

Factor $- 1$ out of $- \sin (a) \sin (b)$ .

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

Step 3.2

Factor $- 1$ out of $\cos (a) \cos (b)$ .

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

Step 3.3

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 3.4

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 3.5

Move the negative in front of the fraction.

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

Step 4

Now consider the right side of the equation.

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

Step 5

Convert to sines and cosines.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Subtract fractions.

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Step 6.1

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

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

Step 6.2

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

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

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