Use de moivres theorem to show that tan 3theta=something

[blueyellow]

Homework Statement

use de moivres theorem to show that
tan 3theta=(3tan theta -tan^3 theta)/(1-3tan^2 theta)

Homework Equations

The Attempt at a Solution

tan 3theta=sin3theta/cos3theta
=(3sin theta - 4 sin^3theta)/(4cos^3theta -3cos theta)

so then i divided everything by cos, and som of the stuff turned into tans, but i can't get it all to turn into tans

 

[Mark44]

Homework Statement

use de moivres theorem to show that
tan 3theta=(3tan theta -tan^3 theta)/(1-3tan^2 theta)

Homework Equations

The Attempt at a Solution

tan 3theta=sin3theta/cos3theta
=(3sin theta - 4 sin^3theta)/(4cos^3theta -3cos theta)

so then i divided everything by cos, and som of the stuff turned into tans, but i can't get it all to turn into tans

I don't see how de Moivre's Formula enters into this problem. It seems to me to be a straightforward exercise in proving a trig identity. Although it can be done by converting the tan terms to sine/cosine, it's probably easier to leave things in terms of the tangent.

The identities that you will need are:
tan(\alpha + \beta) = \frac{tan(\alpha) + tan(\beta)}{1 - tan(\alpha)tan(\beta)}
tan(2\alpha) = \frac{2tan(\alpha)}{1 - tan^2(\alpha)}

 

[rock.freak667]

If you really have to use De Moivre's theorem just consider

(cosθ + isinθ)³

 

[blueyellow]

If you really have to use De Moivre's theorem just consider

(cosθ + isinθ)³

but that's exactly what i considered and by equating real and imaginary coefficients, i ended up with what i wrote in my first post
and then i don't know how to proceed
any help from anyone would b much appreciated

 

[blueyellow]

wel, i emailed the lecturer and was told to just divide top and bottom by cos^3 theta
still don't see how that gets rid of the cos's tho
if u hav any idea, pls speak
thanks

 

[blueyellow]

[Im(cos theta + i sin theta)^3]/[Re(cos theta + i sin theta)^3]

=[3cos^2 theta sin theta - sin^3 theta]/[cos^3 theta - 3 cos theta sin^2 theta]

but how exactly did they go from one step to the next? and how did they get rid of the 'Im' and 'Re'?

 

[SammyS]

Does it help that (cos(θ)+ i sin(θ))³ = (cos(3θ)+ i sin(3θ)) ?