Trigonometry Examples

$\frac{\cos (x) - \sin (x)}{\sin (x) \cos (x)} = \csc (x) - \sec (x)$

Step 1

Start on the left side.

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

Step 2

Reorder terms.

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

Step 3

Now consider the right side of the equation.

$\csc (x) - \sec (x)$

Step 4

Convert to sines and cosines.

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Step 4.1

Apply the reciprocal identity to $\csc (x)$ .

$\frac{1}{\sin (x)} - \sec (x)$

Step 4.2

Apply the reciprocal identity to $\sec (x)$ .

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

Step 5

Subtract fractions.

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Step 5.1

To write $\frac{1}{\sin (x)}$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

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

Step 5.2

To write $- \frac{1}{\cos (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 5.3

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 5.3.1

Multiply $\frac{1}{\sin (x)}$ by $\frac{\cos (x)}{\cos (x)}$ .

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

Step 5.3.2

Multiply $\frac{1}{\cos (x)}$ by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 5.3.3

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

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

Step 5.4

Combine the numerators over the common denominator.

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

Step 6

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

$\frac{\cos (x) - \sin (x)}{\sin (x) \cos (x)} = \csc (x) - \sec (x)$ is an identity