Convert to Rectangular (cos(o)+isin(o))^2

${(\cos (o) + i \sin (o))}^{2}$

Step 1

Rewrite ${(\cos (o) + i \sin (o))}^{2}$ as $(\cos (o) + i \sin (o)) (\cos (o) + i \sin (o))$ .

$(\cos (o) + i \sin (o)) (\cos (o) + i \sin (o))$

Step 2

Expand $(\cos (o) + i \sin (o)) (\cos (o) + i \sin (o))$ using the FOIL Method.

Apply the distributive property.

$\cos (o) (\cos (o) + i \sin (o)) + i \sin (o) (\cos (o) + i \sin (o))$

Apply the distributive property.

$\cos (o) \cos (o) + \cos (o) (i \sin (o)) + i \sin (o) (\cos (o) + i \sin (o))$

Apply the distributive property.

$\cos (o) \cos (o) + \cos (o) (i \sin (o)) + i \sin (o) \cos (o) + i \sin (o) (i \sin (o))$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $\cos (o) \cos (o)$ .

Raise $\cos (o)$ to the power of $1$ .

$\cos^{1} (o) \cos (o) + \cos (o) (i \sin (o)) + i \sin (o) \cos (o) + i \sin (o) (i \sin (o))$

Raise $\cos (o)$ to the power of $1$ .

$\cos^{1} (o) \cos^{1} (o) + \cos (o) (i \sin (o)) + i \sin (o) \cos (o) + i \sin (o) (i \sin (o))$

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

${\cos (o)}^{1 + 1} + \cos (o) (i \sin (o)) + i \sin (o) \cos (o) + i \sin (o) (i \sin (o))$

Add $1$ and $1$ .

$\cos^{2} (o) + \cos (o) (i \sin (o)) + i \sin (o) \cos (o) + i \sin (o) (i \sin (o))$

Rewrite using the commutative property of multiplication.

$\cos^{2} (o) + i \cos (o) \sin (o) + i \sin (o) \cos (o) + i \sin (o) (i \sin (o))$

Multiply $i \sin (o) (i \sin (o))$ .

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

$\cos^{2} (o) + i \cos (o) \sin (o) + i \sin (o) \cos (o) + i^{1} i \sin (o) \sin (o)$

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

$\cos^{2} (o) + i \cos (o) \sin (o) + i \sin (o) \cos (o) + i^{1} i^{1} \sin (o) \sin (o)$

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

$\cos^{2} (o) + i \cos (o) \sin (o) + i \sin (o) \cos (o) + i^{1 + 1} \sin (o) \sin (o)$

Add $1$ and $1$ .

$\cos^{2} (o) + i \cos (o) \sin (o) + i \sin (o) \cos (o) + i^{2} \sin (o) \sin (o)$

Raise $\sin (o)$ to the power of $1$ .

$\cos^{2} (o) + i \cos (o) \sin (o) + i \sin (o) \cos (o) + i^{2} (\sin^{1} (o) \sin (o))$

Raise $\sin (o)$ to the power of $1$ .

$\cos^{2} (o) + i \cos (o) \sin (o) + i \sin (o) \cos (o) + i^{2} (\sin^{1} (o) \sin^{1} (o))$

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

$\cos^{2} (o) + i \cos (o) \sin (o) + i \sin (o) \cos (o) + i^{2} {\sin (o)}^{1 + 1}$

Add $1$ and $1$ .

$\cos^{2} (o) + i \cos (o) \sin (o) + i \sin (o) \cos (o) + i^{2} \sin^{2} (o)$

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

$\cos^{2} (o) + i \cos (o) \sin (o) + i \sin (o) \cos (o) - 1 \sin^{2} (o)$

Rewrite $- 1 \sin^{2} (o)$ as $- \sin^{2} (o)$ .

$\cos^{2} (o) + i \cos (o) \sin (o) + i \sin (o) \cos (o) - \sin^{2} (o)$

Reorder the factors of $i \sin (o) \cos (o)$ .

$\cos^{2} (o) + i \cos (o) \sin (o) + i \cos (o) \sin (o) - \sin^{2} (o)$

Add $i \cos (o) \sin (o)$ and $i \cos (o) \sin (o)$ .

$\cos^{2} (o) + 2 i \cos (o) \sin (o) - \sin^{2} (o)$

Step 4

Move $- \sin^{2} (o)$ .

$\cos^{2} (o) - \sin^{2} (o) + 2 i \cos (o) \sin (o)$

Step 5

Apply the cosine double-angle identity.

$\cos (2 o) + 2 i \cos (o) \sin (o)$

Step 6

Simplify each term.

Add parentheses.

$\cos (2 o) + 2 i (\cos (o) \sin (o))$

Reorder $2 i$ and $\cos (o) \sin (o)$ .

$\cos (2 o) + \cos (o) \sin (o) (2 i)$

Add parentheses.

$\cos (2 o) + \cos (o) (\sin (o) \cdot 2) i$

Reorder $\cos (o)$ and $\sin (o) \cdot 2$ .

$\cos (2 o) + \sin (o) \cdot 2 \cos (o) i$

Reorder $\sin (o)$ and $2$ .

$\cos (2 o) + 2 \cdot \sin (o) \cos (o) i$

Apply the sine double-angle identity.

$\cos (2 o) + \sin (2 o) i$

Step 7

Reorder factors in $\cos (2 o) + \sin (2 o) i$ .

$\cos (2 o) + i \sin (2 o)$

Convert to Rectangular (cos(o)+isin(o))^2

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