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

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