Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 1.4

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

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 1.5

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

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

Step 2

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

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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