Verified answer

Using the trig identity of tan(Θ+Ω)= (tanΘ + tanΩ) / (1 - tanΘtanΩ) and the fact that tan120° = -√3 & tan240° = √3, we have

tan(x + 120°) = (tan x - √3) / (1 + √3tan x)

tan(x + 240°) = (tan x + √3) / (1 - √3tan x)

When we combine everything on the left-hand side (LHS) over the same denominator, we get

tan x + tan(x +120°) + tan(x + 240°) = (9tan x - 3tan^3 x) / (1 - 3tan^3 x)

and so the LHS reduces to (9tan x - 3tan^3 x) / (1 - 3tan²x)

Now rewriting tan3x on the right-hand side (RHS) as tan(x + 2x) and applying the same trig identity...

tan(x + 2x) = (tan x + tan2x) / (1 - tanxtan2x)

Further decomposing the tan2x in the above as as tan(x +x),

tan(x + 2x) = [tanx + 2tanx/(1-tan²x)]/[1 - 2tan²x/(1-tan²x)]

Multiplying both numerator & denominator of the RHS expression by (1-tan²x) gives us (3tanx - tan^3 x) / (1 - 3tan²x)

Multiplying this result by 3 will give us the final expression for the RHS:

3 x (3tanx - tan^3 x) / (1 - 3tan²x) = (9tanx - 3tan^3 x) / (1 - 3tan²x)

So the RHS reduces to (9tanx - 3tan^3 x) / (1 - 3tan²x)

Comparing the LHS and RHS expressions, we see that they are equal and so the equation

tan x + tan(x +120°) + tan(x + 240°) = 3tan(3x)

holds true!

we've tan(P + Q) = (tan P + tan Q)/(a million - tan P*tan Q) employing this. tan(x + one hundred twenty) = (tan x + tan one hundred twenty)/(a million - tan x*tan120) {tan one hundred twenty = tan(one hundred eighty-60) = - tan 60 = -(sqrt3)} So, tan(x + one hundred twenty) = {tan x - (sqrt3)}/{a million + (sqrt3)*tan x} further tan(x + 240) = {tan x + (sqrt3)}/{a million - (sqrt3)*tan x} [tan240 = tan(one hundred eighty + 60) = tan 60] Substituting those in LHS, we get, LHS as = tan x + {tan x - (sqrt3)}/{a million + (sqrt3)*tan x} + + {tan x + (sqrt3)}/{a million - (sqrt3)*tan x} it extremely is = [tan x {a million - 3 tan(x^2)} + {tan x - (sqrt3)}*{a million - (sqrt3)*tan x} + {tan x + (sqrt3)}*{a million + (sqrt3)*tan x}] / {a million - 3 tan(x^2)} it extremely is ={ tan x - tan(x^3)} + {tan x + tan 3x} + {tan x + tan 3x}/ {a million - 3 tan(x^2)} it extremely is = 3{3tan x - tan(x^3)}/{a million - 3 tan(x^2)} = 3tan(3x) [tan 3x = {3tan x - tan(x^3)}/{a million - 3 tan(x^2)}]