Prove cos^3(x) = 3cos(x)+cos(3x)/4?

Question

Thanks

Answer 1

I think you need parentheses in the above for it to be true. Anyhow to show something like this, start with cos3x and use the rule cos(x+y) = cosxcosy - sinxsiny

Using this you get
cos3x =
cos2xcosx - sin2xsinx =
(2(cosx)^2 - 1)cosx - 2cosx(sinx)^2 =
2(cosx)^3 - cosx - 2cosx + 2(cosx)^3 =
4(cosx)^3 - 3cosx

So this gives you:

(cosx)^3 = (3cosx + cos3x)/4

Answer 2

Try it with pi/4 and see what you get. It only has to not work at 1 point to make it untrue.

Answer 3

I tried to prove it, but I think it lacked some salt to my taste.