De Moivre's Theorem for proving

Question

I have been asked by my lecturer to answer this question but I'm having problems doing so. The question is:

Prove that

$$\cos (5\theta) = 16\cos^5\theta - 20\cos^3\theta + 5 \cos\theta\text{.}$$

Answer 1

Hint

$$cos 5\theta = \Re (cos 5\theta + i sin 5\theta)$$

Using De Movire

$$=\Re \left( (cos \theta + i sin \theta)^5\right)$$

Now all you have to do is use binomial expansion and then removing the imaginary parts.