Trigonometry Examples

${(x + i)}^{2} - {(x - i)}^{2}$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(x + i)}^{2}$ as $(x + i) (x + i)$ .

$(x + i) (x + i) - {(x - i)}^{2}$

Step 1.2

Expand $(x + i) (x + i)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$x (x + i) + i (x + i) - {(x - i)}^{2}$

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x i + i x + i i - {(x - i)}^{2}$

Step 1.3.1.2

Multiply $i i$ .

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Step 1.3.1.2.1

Raise $i$ to the power of $1$ .

$x^{2} + x i + i x + i^{1} i - {(x - i)}^{2}$

Step 1.3.1.2.2

Raise $i$ to the power of $1$ .

$x^{2} + x i + i x + i^{1} i^{1} - {(x - i)}^{2}$

Step 1.3.1.2.3

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

$x^{2} + x i + i x + i^{1 + 1} - {(x - i)}^{2}$

Step 1.3.1.2.4

Add $1$ and $1$ .

$x^{2} + x i + i x + i^{2} - {(x - i)}^{2}$

Step 1.3.1.3

Rewrite $i^{2}$ as $- 1$ .

$x^{2} + x i + i x - 1 - {(x - i)}^{2}$

Step 1.3.2

Add $x i$ and $i x$ .

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Step 1.3.2.1

Reorder $i$ and $x$ .

$x^{2} + x i + x i - 1 - {(x - i)}^{2}$

Step 1.3.2.2

Add $x i$ and $x i$ .

$x^{2} + 2 x i - 1 - {(x - i)}^{2}$

Step 1.4

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

$x^{2} + 2 x i - 1 - ((x - i) (x - i))$

Step 1.5

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

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Step 1.5.1

Apply the distributive property.

$x^{2} + 2 x i - 1 - (x (x - i) - i (x - i))$

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

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

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $x$ by $x$ .

$x^{2} + 2 x i - 1 - (x^{2} + x (- i) - i x - i (- i))$

Step 1.6.1.2

Multiply $- i (- i)$ .

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Step 1.6.1.2.1

Multiply $- 1$ by $- 1$ .

$x^{2} + 2 x i - 1 - (x^{2} + x (- i) - i x + 1 i i)$

Step 1.6.1.2.2

Raise $i$ to the power of $1$ .

$x^{2} + 2 x i - 1 - (x^{2} + x (- i) - i x + 1 (i^{1} i))$

Step 1.6.1.2.3

Raise $i$ to the power of $1$ .

$x^{2} + 2 x i - 1 - (x^{2} + x (- i) - i x + 1 (i^{1} i^{1}))$

Step 1.6.1.2.4

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

$x^{2} + 2 x i - 1 - (x^{2} + x (- i) - i x + 1 i^{1 + 1})$

Step 1.6.1.2.5

Add $1$ and $1$ .

$x^{2} + 2 x i - 1 - (x^{2} + x (- i) - i x + 1 i^{2})$

Step 1.6.1.2.6

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

$x^{2} + 2 x i - 1 - (x^{2} + x (- i) - i x + i^{2})$

Step 1.6.1.3

Rewrite $i^{2}$ as $- 1$ .

$x^{2} + 2 x i - 1 - (x^{2} + x (- i) - i x - 1)$

Step 1.6.2

Move $i$ .

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

Step 1.6.3

Subtract $x i$ from $- 1 \cdot x i$ .

$x^{2} + 2 x i - 1 - (x^{2} - 2 x i - 1)$

Step 1.7

Apply the distributive property.

$x^{2} + 2 x i - 1 - x^{2} - (- 2 x i) - - 1$

Step 1.8

Simplify.

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Step 1.8.1

Multiply $- 2$ by $- 1$ .

$x^{2} + 2 x i - 1 - x^{2} + 2 (x i) - - 1$

Step 1.8.2

Multiply $- 1$ by $- 1$ .

$x^{2} + 2 x i - 1 - x^{2} + 2 (x i) + 1$

$x^{2} + 2 x i - 1 - x^{2} + 2 x i + 1$

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $x^{2} + 2 x i - 1 - x^{2} + 2 x i + 1$ .

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Step 2.1.1

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

$2 x i - 1 + 0 + 2 x i + 1$

Step 2.1.2

Add $2 x i - 1$ and $0$ .

$2 x i - 1 + 2 x i + 1$

Step 2.1.3

Add $- 1$ and $1$ .

$2 x i + 2 x i + 0$

Step 2.1.4

Add $2 x i + 2 x i$ and $0$ .

$2 x i + 2 x i$

Step 2.2

Add $2 x i$ and $2 x i$ .

$4 x i$