Trigonometry Examples

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

Step 1

Rewrite ${(x - y)}^{2}$ as $(x - y) (x - y)$ .

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.3

Rewrite using the commutative property of multiplication.

$x^{2} - x y - y x - 1 \cdot - 1 y \cdot y = x^{2} - 2 x y + y^{2}$

Step 3.1.4

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

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Step 3.1.4.1

Move $y$ .

$x^{2} - x y - y x - 1 \cdot - 1 (y \cdot y) = x^{2} - 2 x y + y^{2}$

Step 3.1.4.2

Multiply $y$ by $y$ .

$x^{2} - x y - y x - 1 \cdot - 1 y^{2} = x^{2} - 2 x y + y^{2}$

Step 3.1.5

Multiply $- 1$ by $- 1$ .

$x^{2} - x y - y x + 1 y^{2} = x^{2} - 2 x y + y^{2}$

Step 3.1.6

Multiply $y^{2}$ by $1$ .

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

Step 3.2

Subtract $y x$ from $- x y$ .

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Step 3.2.1

Move $y$ .

$x^{2} - x y - 1 x y + y^{2} = x^{2} - 2 x y + y^{2}$

Step 3.2.2

Subtract $x y$ from $- x y$ .

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

Step 4

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

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