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

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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

ctgx = 3. To find ctg(2x).

Solution:

So cotx = 3 implies tanx =1/3.

tan2x = 2tanx/(1-(tanx)^2). Or

ctg(2x) = (1-(tanx)^2)/(2tanx) = [1-(1/3)^2]/(2(1/3)) = 3(9-1)/(2*9 )= 8/6 = 4/3

ctg2x=ctg(x+x)

ctg(x+x)=1/tg(x+x)

1/tg(x+x)=[1-(tgx)^2]/2tgx=[1-(1/ctgx)^2]/2*1/ctgx

ctg2x=(ctg x)/2[1-1/(ctgx)^2]=3/[2(1-1/9)]=(3/2)(8/9)=4/3

**ctg2x=4/3**