Find out ctg(2x) if ctg(x) = 3

we need to determine ctg(2x)  if we know that ctg(x)=3

we know that ctg(2x)=1/tan(2x) =1/tan(x+x)

We know that tan(x+y)= (tanx+tany)/1-tanx*tany)

==> 1/tan(x+x)= [1-(tanx)^2]/2tanx

 We know that ctg=3 , then tanx= 1/3

Substitute:

ctg(2x)= 1-(1/3)^2 / 2(1/3)

          = (1-1/9) / (2/3)

          = 8(3)/9(2)= 8/6= 4/3