Trigonometric Identities - Assignment

Question

How do I simplify Cos(5 theta)?

I got as far as Cos(2theta + 3theta). Do I then say Cos(2theta + 3theta) = Cos(2theta) + Cos(3theta)?

In that case, how do I get Cos(3theta)?

Answer 1

$$\cos(A+B)=\cos A\cos B-\sin A\sin B\ne \cos A+\cos B\text{ (in general)}$$

So, $$\cos5\theta=\cos(2\theta+3\theta)=\cos2\theta\cos3\theta-\sin2\theta\sin3\theta$$

$$=(2\cos^2\theta-1)(4\cos^3\theta-3\cos\theta)-2\sin\theta\cos\theta(3\sin\theta-4\sin^3\theta)$$

$$=(2\cos^2\theta-1)(4\cos^3\theta-3\cos\theta)-2\sin^2\theta\cos\theta(3-4\sin^2\theta)$$

Use $\sin^2\theta=1-\cos^2\theta$ and simplify

Answer 2

$$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ first use 5$\theta$=3$\theta$ + 2$\theta$ then 3$\theta$=2$\theta$+$\theta$ and 2$\theta$=$\theta$+$\theta$ ;

you'll get higher powers but don't lose hope.