Trigonometry Examples

$(\csc (x) - \sec (x)) (\cos (x) + \sin (x))$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 2

Expand $(\frac{1}{\sin (x)} - \frac{1}{\cos (x)}) (\cos (x) + \sin (x))$ using the FOIL Method.

Tap for more steps...

Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Combine $\frac{1}{\sin (x)}$ and $\cos (x)$ .

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

Step 3.1.2

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

Tap for more steps...

Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

Rewrite the expression.

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

Step 3.1.3

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

Tap for more steps...

Step 3.1.3.1

Move the leading negative in $- \frac{1}{\cos (x)}$ into the numerator.

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

Step 3.1.3.2

Cancel the common factor.

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

Step 3.1.3.3

Rewrite the expression.

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

Step 3.1.4

Combine $\sin (x)$ and $\frac{1}{\cos (x)}$ .

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

Step 3.2

Subtract $1$ from $1$ .

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

Step 3.3

Add $\frac{\cos (x)}{\sin (x)}$ and $0$ .

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

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 4.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\cot (x) - \tan (x)$