Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 2

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

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

Step 3

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

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

Step 4

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 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $\cos (x)$ by $1$ .

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

Step 6.3

Cancel the common factor of $\cos (x)$ .

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Step 6.3.1

Cancel the common factor.

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

Step 6.3.2

Rewrite the expression.

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

Step 6.4

Subtract $\sin (x)$ from $\sin (x)$ .

$\frac{\cos (x) + 0}{\cos (x) \sin (x)}$

Step 6.5

Add $\cos (x)$ and $0$ .

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

Step 7

Cancel the common factor of $\cos (x)$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\csc (x)$