Well... Sage does not, indeed, "automagically" apply all possible trig 
identities <http://www2.clarku.edu/faculty/djoyce/trig/identities.html>. 
But perusing them allow to select (with a tiny grain of salt) the right one:

sage: tan(1/2*arctan(12/5)).subs(tan(w0/2)==sin(w0)/(1+cos(w0)))
2/3

HTH,


Le mercredi 20 février 2019 09:15:59 UTC+1, Michael Beeson a écrit :
>
> sage: t
>
> tan(1/2*arctan(12/5))
>
> sage: t.trig_simplify()
>
> sin(1/2*arctan(12/5))/cos(1/2*arctan(12/5))
>
> sage: n(t)
>
> 0.666666666666667
>
>
> But trig_simplify  couldn't get 2/3.  Maybe there
>
> is a fancier command that will do it?
>
>   
>

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