Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply the difference 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

Now consider the right side of the equation.

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

Step 4

Convert to sines and cosines.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

Add fractions.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

Multiply $\frac{\sin (a)}{\cos (a)}$ by $\frac{\sin (b)}{\sin (b)}$ .

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

Step 5.3.2

Multiply $\frac{\cos (b)}{\sin (b)}$ by $\frac{\cos (a)}{\cos (a)}$ .

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

Step 5.3.3

Reorder the factors of $\sin (b) \cos (a)$ .

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

Step 5.4

Combine the numerators over the common denominator.

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

Step 6

Reorder terms.

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

Step 7

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

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