Trigonometry Examples

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

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

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

Subtract $1$ from $1$ .

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

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

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

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

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

Combine exponents.

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Raise $\sin (x)$ to the power of $1$ .

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

Raise $\sin (x)$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Move the negative in front of the fraction.

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