Trigonometry Examples

$(a - b) {(a^{- 1} b + a b^{- 1})}^{- 1}$

Step 1

Simplify each term.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(a - b) {(\frac{1}{a} b + a b^{- 1})}^{- 1}$

Step 1.2

Combine $\frac{1}{a}$ and $b$ .

$(a - b) {(\frac{b}{a} + a b^{- 1})}^{- 1}$

Step 1.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(a - b) {(\frac{b}{a} + a \frac{1}{b})}^{- 1}$

Step 1.4

Combine $a$ and $\frac{1}{b}$ .

$(a - b) {(\frac{b}{a} + \frac{a}{b})}^{- 1}$

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(a - b) \frac{1}{\frac{b}{a} + \frac{a}{b}}$

Step 3

Simplify the denominator.

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Step 3.1

To write $\frac{b}{a}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

$(a - b) \frac{1}{\frac{b}{a} \cdot \frac{b}{b} + \frac{a}{b}}$

Step 3.2

To write $\frac{a}{b}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$(a - b) \frac{1}{\frac{b}{a} \cdot \frac{b}{b} + \frac{a}{b} \cdot \frac{a}{a}}$

Step 3.3

Write each expression with a common denominator of $a b$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{b}{a}$ by $\frac{b}{b}$ .

$(a - b) \frac{1}{\frac{b \cdot b}{a b} + \frac{a}{b} \cdot \frac{a}{a}}$

Step 3.3.2

Multiply $\frac{a}{b}$ by $\frac{a}{a}$ .

$(a - b) \frac{1}{\frac{b \cdot b}{a b} + \frac{a \cdot a}{b a}}$

Step 3.3.3

Reorder the factors of $b a$ .

$(a - b) \frac{1}{\frac{b \cdot b}{a b} + \frac{a \cdot a}{a b}}$

Step 3.4

Combine the numerators over the common denominator.

$(a - b) \frac{1}{\frac{b \cdot b + a \cdot a}{a b}}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $b$ by $b$ .

$(a - b) \frac{1}{\frac{b^{2} + a \cdot a}{a b}}$

Step 3.5.2

Multiply $a$ by $a$ .

$(a - b) \frac{1}{\frac{b^{2} + a^{2}}{a b}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$(a - b) (1 \frac{a b}{b^{2} + a^{2}})$

Step 5

Multiply $\frac{a b}{b^{2} + a^{2}}$ by $1$ .

$(a - b) \frac{a b}{b^{2} + a^{2}}$

Step 6

Multiply $a - b$ by $\frac{a b}{b^{2} + a^{2}}$ .

$\frac{(a - b) (a b)}{b^{2} + a^{2}}$

Step 7

Reorder factors in $\frac{(a - b) a b}{b^{2} + a^{2}}$ .

$\frac{a b (a - b)}{b^{2} + a^{2}}$