Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite $\frac{\sin (x - \cos (x))}{\sin (x)}$ as a product.

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

Step 1.2

Write $\sin (x - \cos (x))$ as a fraction with denominator $1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Divide $\sin (x - \cos (x))$ by $1$ .

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

Step 1.3.2

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

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

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

Step 2

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

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

Step 3

Combine $- \sin (x - \cos (x)) \csc (x)$ and $\frac{\cos (x)}{\cos (x)}$ .

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

Step 4

Combine the numerators over the common denominator.

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