Recurrence relation of tangent

[davedave]

Is it possible to find a recurrence relation of tan(nx) where n is a positive integer and x is a real variable?

My friend said that it is possible.

I don't see how to do it.

Does anyone have a way to do it?

 

[HallsofIvy]

You should know the sum formula tan(A+ B)= (tan(A)+ tan(B))/(1+ tan(A)(tan(B)).

Using that, tan((n+1)x)= tan(x+ nx= (tan(x)+ tan(nx))/(1+ tan(x)tan(nx))
or, if we write T_n= tan(nx),

T_n+1= (T_1+ T_n)/(1+ T_1T_n), a recurrance relation.