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

Expand the right hand side of using the binomial theorem.

Expand:

Use the Binomial Theorem.

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 .

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 .

Factor out .

Factor out .

Rewrite as .

Rewrite as .

Rewrite as .

Raise to the power of .

Multiply by .

Rewrite as .

Rewrite as .

Reorder factors in .

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

Expand Using DeMoivre’s Theorem sin(7x)