Trigonometry Examples

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

Step 1

Start on the right side.

$\cot (x) + \tan (x)$

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.

$\frac{\cos (x)}{\sin (x)} + \tan (x)$

Step 2.2

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

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

Step 3

Add fractions.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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

Step 5

Apply Pythagorean identity.

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Step 5.1

Rearrange terms.

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

Step 5.2

Apply pythagorean identity.

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

Step 6

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

$\csc (x) \cdot \sec (x)$

Step 7

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

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