Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Write $1$ as a fraction with a common denominator.

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

Step 3.1.2

Combine the numerators over the common denominator.

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

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3

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

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

Step 4

Now consider the right side of the equation.

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

Step 5

Convert to sines and cosines.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Subtract fractions.

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Step 6.1

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 6.2

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 6.3

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 6.3.1

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 6.3.2

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 6.3.3

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

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

Step 6.4

Combine the numerators over the common denominator.

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

Step 7

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

$\frac{1 - \tan (x)}{\sin (x)} = \csc (x) - \sec (x)$ is an identity