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.