Using e^ix to determine a trig identity

[josephcollins]

Hi people, could someone help me with this

Q. Write cosx and sinx in terms of e^ix and e^-ix respectively

So I wrote that cosx=Re(e^ix)=Re(e^-ix)

and sinx =Im(e^ix) and -Im(e^ix)

I think the above identites are correct, now I must use this to show that

16cos^3(x)sin^2(x) = 2cosx - cos3x - cos5x

 

[Zurtex]

$$e^{ix} = \cos x + i \sin x$$

$$e^{-ix} = \cos x - i \sin x$$

Look at the two above and think how you could rearrange them so you have one for sin(x) in terms of e^(ix) and e^(-ix) and same for cos(x).

 

[Dr Transport]

also remember, $$e^{inx} = \cos(nx) + i\sin(nx) = (e^{ix})^n$$. This is all you need to find any trig identity...