Simplify (cos(x))/(1+sin(x))-(cos(x))/(1-sin(x))

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

Step 1

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $(1 + \sin (x)) (1 - \sin (x))$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $(1 - \sin (x)) (1 + \sin (x))$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Factor $\cos (x)$ out of $\cos (x) (1 - \sin (x)) - \cos (x) (1 + \sin (x))$ .

Factor $\cos (x)$ out of $- \cos (x) (1 + \sin (x))$ .

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

Factor $\cos (x)$ out of $\cos (x) (1 - \sin (x)) + \cos (x) (- (1 + \sin (x)))$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

Subtract $1$ from $1$ .

$\frac{\cos (x) (0 - \sin (x) - \sin (x))}{(1 + \sin (x)) (1 - \sin (x))}$

Subtract $\sin (x)$ from $0$ .

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

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

$\frac{\cos (x) \cdot - 2 \sin (x)}{(1 + \sin (x)) (1 - \sin (x))}$

Step 6

Simplify the expression.

Move $- 2$ to the left of $\cos (x)$ .

$\frac{- 2 \cdot \cos (x) \sin (x)}{(1 + \sin (x)) (1 - \sin (x))}$

Move the negative in front of the fraction.

$- \frac{2 \cos (x) \sin (x)}{(1 + \sin (x)) (1 - \sin (x))}$