Simplify (sec(theta)-csc(theta))(cos(theta)+sin(theta))

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

Step 1

Simplify each term.

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

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

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

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Subtract $1$ from $1$ .

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

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

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

Step 4

Simplify each term.

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

$\tan (θ) - \frac{\cos (θ)}{\sin (θ)}$

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

$\tan (θ) - \cot (θ)$