Rewrite 2 tan 3x in terms of tan x

Question

Rewrite 2 tan 3x in terms of tan x

Answer 1

First, recognize that 3x=2x+x

Thus tan(3x)=tan(2x+x)

By the sum formula, tan(2x+x)=[tan(2x)+tan(x)]/[1-tan(2x)tan(x)]

By tangent double angle formula, tan(2x)=2tan(x)/(1-tan²(x))

Substituting that in we get [2tan(x)/(1-tan²(x))+tan(x)]/[1-2tan(x)/(1-tan²(x))tan(x))]

Multiplying top and bottom by the quantity 1-tan²(x) to get rid of denominators gives

[2tan(x)+tan(x)(1-tan²(x))]/[1-tan²(x)-2tan(x)(tan(x))]

Simplifying, we get [2tan(x)+tan(x)-tan³(x)]/[1-tan²(x)-2tan²(x)]

Combining like terms, we get [3tan(x)-tan²(x)]/[1-3tan³(x)]

This is all in terms of tan(x) so this should suffice