(iii) Find the most general function that, for all x and y, satisfies the identity
t(x) + t(y) = t(z)
where z = (x+y)/(1-xy)
I can't think of how to do this. I can see some relation to the t=tan((1/2)A) substitution where sin(A) = (2t)/(1-t^2) but don't know how to use it in this context.
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tan(a+b) = ( tan a + tan b )/(1 - tan a.tan b)
Let t = arctan, x = tan a and y = tan b
and you now have an example of this function t.
Now, the find the most general form, you're probably going to have to have a look at 2-dimensional Taylor series.
t(x) = arctan(x) ? Now I'm confused because I can't see how this satisfies t(x) + t(y) = t(z) still? :S I think it's the manipulation of these general functions which I am messing up on, it's something I'm unfamiliar with..
Thanks so much for your help James.
Nevermind I understand it now, you have no idea how much this is going to help me James haha.