# 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.
Simplify each term.
Apply the product rule to .
Rewrite using the commutative property of multiplication.
Rewrite as .
Multiply by .
Apply the product rule to .
Rewrite using the commutative property of multiplication.
Factor out .
Rewrite as .
Rewrite as .
Multiply by .
Apply the product rule to .
Rewrite using the commutative property of multiplication.
Rewrite as .
Rewrite as .
Rewrite as .
Raise to the power of .
Multiply by .
Apply the product rule to .
Rewrite using the commutative property of multiplication.
Factor out .
Rewrite as .
Rewrite as .
Rewrite as .
Raise to the power of .
Multiply by .
Apply the product rule to .
Rewrite using the commutative property of multiplication.
Factor out .
Rewrite as .
Rewrite as .
Rewrite as .
Raise to the power of .
Multiply by .
Rewrite as .
Multiply by .
Apply the product rule to .
Rewrite as .
Factor out .
Factor out .
Rewrite as .
Rewrite as .
Rewrite as .
Raise to the power of .
Multiply by .
Rewrite as .
Rewrite as .
Multiply by .
Apply the product rule to .
Rewrite as .
Rewrite as .
Rewrite as .
Rewrite as .
Raise to the power of .
One to any power is one.
Multiply by .
Reorder factors in .
Step 5
Take out the expressions with the imaginary part, which are equal to . Remove the imaginary number .