# Trigonometry Examples

Step 1

A good method to expand is by using De Moivre's theorem . When , .

Step 2

Expand the right hand side of using the binomial theorem.

Expand:

Step 3

Use the Binomial Theorem.

Step 4

Step 4.1

Simplify each term.

Step 4.1.1

Apply the product rule to .

Step 4.1.2

Rewrite using the commutative property of multiplication.

Step 4.1.3

Rewrite as .

Step 4.1.4

Multiply by .

Step 4.1.5

Apply the product rule to .

Step 4.1.6

Rewrite using the commutative property of multiplication.

Step 4.1.7

Factor out .

Step 4.1.8

Rewrite as .

Step 4.1.9

Rewrite as .

Step 4.1.10

Multiply by .

Step 4.1.11

Apply the product rule to .

Step 4.1.12

Rewrite using the commutative property of multiplication.

Step 4.1.13

Rewrite as .

Step 4.1.13.1

Rewrite as .

Step 4.1.13.2

Rewrite as .

Step 4.1.13.3

Raise to the power of .

Step 4.1.14

Multiply by .

Step 4.1.15

Apply the product rule to .

Step 4.1.16

Rewrite using the commutative property of multiplication.

Step 4.1.17

Factor out .

Step 4.1.18

Rewrite as .

Step 4.1.18.1

Rewrite as .

Step 4.1.18.2

Rewrite as .

Step 4.1.18.3

Raise to the power of .

Step 4.1.19

Multiply by .

Step 4.1.20

Apply the product rule to .

Step 4.1.21

Factor out .

Step 4.1.22

Rewrite as .

Step 4.1.22.1

Rewrite as .

Step 4.1.22.2

Rewrite as .

Step 4.1.22.3

Raise to the power of .

Step 4.1.23

Multiply by .

Step 4.1.24

Rewrite as .

Step 4.1.25

Rewrite as .

Step 4.1.26

Multiply by .

Step 4.1.27

Apply the product rule to .

Step 4.1.28

Rewrite as .

Step 4.1.28.1

Factor out .

Step 4.1.28.2

Factor out .

Step 4.1.29

Rewrite as .

Step 4.1.29.1

Rewrite as .

Step 4.1.29.2

Rewrite as .

Step 4.1.29.3

Raise to the power of .

Step 4.1.30

Multiply by .

Step 4.1.31

Rewrite as .

Step 4.1.32

Rewrite as .

Step 4.2

Reorder factors in .

Step 5

Take out the expressions with the imaginary part, which are equal to . Remove the imaginary number .