Trigonometry Examples

$\tan (a) + \cot (a) = \sec (a) \cdot \csc (a)$

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Add fractions.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

$\frac{\sin^{2} (a) + \cos^{2} (a)}{\cos (a) \sin (a)}$

Step 5

Apply pythagorean identity.

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

Step 6

Rewrite $\frac{1}{\cos (a) \sin (a)}$ as $\sec (a) \cdot \csc (a)$ .

$\sec (a) \cdot \csc (a)$

Step 7

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

$\tan (a) + \cot (a) = \sec (a) \cdot \csc (a)$ is an identity