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4.40 Simplifying multiple angles in terms of trig powers

Use de Moivre’s Theorem directly by

cos⁡n⁢θ+i⁢sin⁡n⁢θ=ei⁢n⁢θ=(cos⁡θ+i⁢sin⁡θ)n\cos n\theta+i\sin n\theta=e^{in\theta}=(\cos\theta+i\sin\theta)^{n}

and expanding the RHS by the binomial theorem.

4.40 Example

To prove that

cos⁡6⁢θ=32⁢cos6⁡θ-48⁢cos4⁡θ+18⁢cos2⁡θ-1\cos 6\theta=32\cos^{6}\theta-48\cos^{4}\theta+18\cos^{2}\theta-1

and

sin⁡6⁢θ=6⁢cos5⁡θ⁢sin⁡θ-20⁢cos3⁡θ⁢sin3⁡θ+6⁢cos⁡θ⁢sin5⁡θ.\sin 6\theta=6\cos^{5}\theta\sin\theta-20\cos^{3}\theta\sin^{3}\theta+6\cos% \theta\sin^{5}\theta.

Note that we need both cos⁡θ\cos\theta and sin⁡θ\sin\theta in the formula for sin⁡6⁢θ\sin 6\theta.