I have some difficulties in my little example:

K:=(I_tr*sin(a+b)²-I_trsin(a-b)²)/(I_tr*sin(a+b)²+I_trsin(a-b)²)+2*I_ex

... after few transformations ...

K:=sin(2a)sin(2b)/(sin(a)²cos(b)²+cos(a)²sin(b)²+I_ex/I_tr)

also we have:

r:=I_ex/I_tr

d:=1+2*r

My question is in the title, here is simple example: 

K:=(sin^2(a+b)-sin^2(a-b))

I use formula of abridged multiplication (with help of "factor")

>factor(K)

  (sin(a+b) - sin(a-b))(sin(a+b) + sin(a-b))

Then expand:

>expand(%, trig)

  4sin(a)cos(b)cos(a)sin(b)

And all I whant now is to use double angle formula like this:

4sin(a)cos(b)cos(a)sin(b) = sin(2a)sin(2b)