simplification of trigonometric equation

I have an expression on the form

Code:

`a*sin(w)`^{3} + b*sin(w)

-------------------- (1)

c*cos(w)^{3} + d*cos(w)

My goal is to arrive at an expression on this form

Code:

` `

sin(w)^{3} sin(w)

p*--------- + q*------- (2)

cos(w)^{3} cos(w)

I believe it should be possible, I just can't find a way to manipulate (1) in to (2).

Any help is much appreciated!

Br

niaren

Re: simplification of trigonometric equation

Hint :

$\displaystyle \sin^2 w=\frac{\tan^2 w}{1+\tan^2 w} ~\text{and}~ \cos^2 w=\frac{1}{1+\tan^2 w}$

Re: simplification of trigonometric equation

Holy smoke, that was fast!

I got it now.

I'm a little surprised you came up with the hint so fast.

For the equality to hold, d == 0 must be satisfied.

In fact, I'm considering a more general problem of manipulating

Code:

`aN*sin(w)^N + .... + a3*sin(w)^3 + a1*sin(w)`

------------------------------------------------ (1)

bN*cos(w)^N + .....+ b3*cos(w)^3 + b1*cos(w)

into

Code:

` `

sin(w)^N sin(w)^3 sin(w)

cN*--------- + c3 ----------- + c1*-------- (2)

cos(w)^N cos(w)^3 cos(w)

Where N is odd.

I suspect your hint can be used for any N by using appropriate powers of you hints. I'll look into that.