# 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
Simplify terms.
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 .