Trigonometry Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Simplify each term.

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Step 3.1

Multiply $a$ by $a$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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Step 3.3.1

Move $b$ .

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

Step 3.3.2

Multiply $b$ by $b$ .

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

Step 4

Since the two sides have been shown to be equivalent, the equation is an identity.

$a^{2} - b^{2} = (a + b) (a - b)$ is an identity.