Trigonometry Examples

$(a + b) (a - b) = a^{2} - b^{2}$

Step 1

Simplify $(a + b) (a - b)$ .

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Step 1.1

Rewrite.

$0 + 0 + (a + b) (a - b) = a^{2} - b^{2}$

Step 1.2

Simplify by adding zeros.

$(a + b) (a - b) = a^{2} - b^{2}$

Step 1.3

Expand $(a + b) (a - b)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$a (a - b) + b (a - b) = a^{2} - b^{2}$

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

Step 1.4

Simplify terms.

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Step 1.4.1

Combine the opposite terms in $a \cdot a + a (- b) + b a + b (- b)$ .

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Step 1.4.1.1

Reorder the factors in the terms $a (- b)$ and $b a$ .

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

Step 1.4.1.2

Add $- a b$ and $a b$ .

$a \cdot a + 0 + b (- b) = a^{2} - b^{2}$

Step 1.4.1.3

Add $a \cdot a$ and $0$ .

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

Step 1.4.2

Simplify each term.

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Step 1.4.2.1

Multiply $a$ by $a$ .

$a^{2} + b (- b) = a^{2} - b^{2}$

Step 1.4.2.2

Rewrite using the commutative property of multiplication.

$a^{2} - b \cdot b = a^{2} - b^{2}$

Step 1.4.2.3

Multiply $b$ by $b$ by adding the exponents.

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Step 1.4.2.3.1

Move $b$ .

$a^{2} - (b \cdot b) = a^{2} - b^{2}$

Step 1.4.2.3.2

Multiply $b$ by $b$ .

$a^{2} - b^{2} = a^{2} - b^{2}$

Step 2

Move all terms containing $a$ to the left side of the equation.

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Step 2.1

Subtract $a^{2}$ from both sides of the equation.

$a^{2} - b^{2} - a^{2} = - b^{2}$

Step 2.2

Combine the opposite terms in $a^{2} - b^{2} - a^{2}$ .

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Step 2.2.1

Subtract $a^{2}$ from $a^{2}$ .

$- b^{2} + 0 = - b^{2}$

Step 2.2.2

Add $- b^{2}$ and $0$ .

$- b^{2} = - b^{2}$

Step 3

Divide each term in $- b^{2} = - b^{2}$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- b^{2} = - b^{2}$ by $- 1$ .

$\frac{- b^{2}}{- 1} = \frac{- b^{2}}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{b^{2}}{1} = \frac{- b^{2}}{- 1}$

Step 3.2.2

Divide $b^{2}$ by $1$ .

$b^{2} = \frac{- b^{2}}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Dividing two negative values results in a positive value.

$b^{2} = \frac{b^{2}}{1}$

Step 3.3.2

Divide $b^{2}$ by $1$ .

$b^{2} = b^{2}$

Step 4

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| b | = | b |$

Step 5

Solve for $b$ .

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Step 5.1

Rewrite the absolute value equation as four equations without absolute value bars.

$b = b$

$b = - b$

$- b = b$

$- b = - b$

Step 5.2

After simplifying, there are only two unique equations to be solved.

$b = b$

$b = - b$

Step 5.3

Solve $b = b$ for $b$ .

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Step 5.3.1

Move all terms containing $b$ to the left side of the equation.

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Step 5.3.1.1

Subtract $b$ from both sides of the equation.

$b - b = 0$

Step 5.3.1.2

Subtract $b$ from $b$ .

$0 = 0$

Step 5.3.2

Since $0 = 0$ , the equation will always be true.

Always true

Step 5.4

Solve $b = - b$ for $b$ .

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Step 5.4.1

Move all terms containing $b$ to the left side of the equation.

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Step 5.4.1.1

Add $b$ to both sides of the equation.

$b + b = 0$

Step 5.4.1.2

Add $b$ and $b$ .

$2 b = 0$

Step 5.4.2

Divide each term in $2 b = 0$ by $2$ and simplify.

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Step 5.4.2.1

Divide each term in $2 b = 0$ by $2$ .

$\frac{2 b}{2} = \frac{0}{2}$

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

$\frac{2 b}{2} = \frac{0}{2}$

Step 5.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0}{2}$

Step 5.4.2.3

Simplify the right side.

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Step 5.4.2.3.1

Divide $0$ by $2$ .

$b = 0$

Step 5.5

List all of the solutions.

$b = 0$