Trigonometry Examples

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

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

Step 1

To write

\frac{1}{1 + sin (x)}

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

\frac{1 - sin (x)}{1 - sin (x)}

$\frac{1 - \sin (x)}{1 - \sin (x)}$ .

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

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

Step 2

To write

- \frac{1}{1 - sin (x)}

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

\frac{1 + sin (x)}{1 + sin (x)}

$\frac{1 + \sin (x)}{1 + \sin (x)}$ .

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

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

Step 3

Write each expression with a common denominator of

(1 + sin (x)) (1 - sin (x))

$(1 + \sin (x)) (1 - \sin (x))$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 3.1

Multiply

\frac{1}{1 + sin (x)}

$\frac{1}{1 + \sin (x)}$ by

\frac{1 - sin (x)}{1 - sin (x)}

$\frac{1 - \sin (x)}{1 - \sin (x)}$ .

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

Step 3.2

Multiply

$\frac{1}{1 - \sin (x)}$ by

$\frac{1 + \sin (x)}{1 + \sin (x)}$ .

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

Step 3.3

Reorder the factors of

$(1 - \sin (x)) (1 + \sin (x))$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply

$- 1$ by

$1$ .

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

Step 5.3

Subtract

$1$ from

$1$ .

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

Step 5.4

Subtract

$\sin (x)$ from

$0$ .

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

Step 5.5

Subtract

$\sin (x)$ from

$- \sin (x)$ .

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

Step 6

Move the negative in front of the fraction.

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