Trigonometry Examples

$x^{4} - y^{4} = (x - y) (x + y) (x^{2} + y^{2})$

Step 1

Expand $(x - y) (x + y)$ using the FOIL Method.

Tap for more steps...

Step 1.1

Apply the distributive property.

$x^{4} - y^{4} = (x (x + y) - y (x + y)) (x^{2} + y^{2})$

Step 1.2

Apply the distributive property.

$x^{4} - y^{4} = (x \cdot x + x y - y (x + y)) (x^{2} + y^{2})$

Step 1.3

Apply the distributive property.

$x^{4} - y^{4} = (x \cdot x + x y - y x - y \cdot y) (x^{2} + y^{2})$

Step 2

Combine the opposite terms in $x \cdot x + x y - y x - y \cdot y$ .

Tap for more steps...

Step 2.1

Reorder the factors in the terms $x y$ and $- y x$ .

$x^{4} - y^{4} = (x \cdot x + x y - x y - y \cdot y) (x^{2} + y^{2})$

Step 2.2

Subtract $x y$ from $x y$ .

$x^{4} - y^{4} = (x \cdot x + 0 - y \cdot y) (x^{2} + y^{2})$

Step 2.3

Add $x \cdot x$ and $0$ .

$x^{4} - y^{4} = (x \cdot x - y \cdot y) (x^{2} + y^{2})$

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

Multiply $x$ by $x$ .

$x^{4} - y^{4} = (x^{2} - y \cdot y) (x^{2} + y^{2})$

Step 3.2

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 3.2.1

Move $y$ .

$x^{4} - y^{4} = (x^{2} - (y \cdot y)) (x^{2} + y^{2})$

Step 3.2.2

Multiply $y$ by $y$ .

$x^{4} - y^{4} = (x^{2} - y^{2}) (x^{2} + y^{2})$

Step 4

Expand $(x^{2} - y^{2}) (x^{2} + y^{2})$ using the FOIL Method.

Tap for more steps...

Step 4.1

Apply the distributive property.

$x^{4} - y^{4} = x^{2} (x^{2} + y^{2}) - y^{2} (x^{2} + y^{2})$

Step 4.2

Apply the distributive property.

$x^{4} - y^{4} = x^{2} x^{2} + x^{2} y^{2} - y^{2} (x^{2} + y^{2})$

Step 4.3

Apply the distributive property.

$x^{4} - y^{4} = x^{2} x^{2} + x^{2} y^{2} - y^{2} x^{2} - y^{2} y^{2}$

Step 5

Combine the opposite terms in $x^{2} x^{2} + x^{2} y^{2} - y^{2} x^{2} - y^{2} y^{2}$ .

Tap for more steps...

Step 5.1

Reorder the factors in the terms $x^{2} y^{2}$ and $- y^{2} x^{2}$ .

$x^{4} - y^{4} = x^{2} x^{2} + x^{2} y^{2} - x^{2} y^{2} - y^{2} y^{2}$

Step 5.2

Subtract $x^{2} y^{2}$ from $x^{2} y^{2}$ .

$x^{4} - y^{4} = x^{2} x^{2} + 0 - y^{2} y^{2}$

Step 5.3

Add $x^{2} x^{2}$ and $0$ .

$x^{4} - y^{4} = x^{2} x^{2} - y^{2} y^{2}$

Step 6

Simplify each term.

Tap for more steps...

Step 6.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 6.1.1

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

$x^{4} - y^{4} = x^{2 + 2} - y^{2} y^{2}$

Step 6.1.2

Add $2$ and $2$ .

$x^{4} - y^{4} = x^{4} - y^{2} y^{2}$

Step 6.2

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

Tap for more steps...

Step 6.2.1

Move $y^{2}$ .

$x^{4} - y^{4} = x^{4} - (y^{2} y^{2})$

Step 6.2.2

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

$x^{4} - y^{4} = x^{4} - y^{2 + 2}$

Step 6.2.3

Add $2$ and $2$ .

$x^{4} - y^{4} = x^{4} - y^{4}$

Step 7

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

$x^{4} - y^{4} = (x - y) (x + y) (x^{2} + y^{2})$ is an identity.