Question

if tan x+tan(x+π/3)+tan(x+2π/3)=3,show that tan3x=3x=1l​

Answer 1

Answer:

100

Step-by-step explanation:

tan x+tan(x+π/3)+tan(x+2π/3)=3

//Remember Tan(A+B) = TanA + TanB/ 1 - TanATanB

tanx + [tanx + tanπ/3/1 - tanxtanπ/3] + [tanx + tan2π/3/1 - tanxtan2π/3] = 3

tanx + [tanx + √3 / 1 - √3tanx] + [tanx - √3 / 1 + √3tanx] = 3

tanx +[(tanx + √3) (1+ √3tanx) + (tanx - √3)(1 - √3tanx)/ (1 - √3tanx)(1 + √3tanx] = 3

tanx + (tanx + √3tan²x + √3 + 3tanx + tanx - √3tan²x - √3 + 3tanx}/{1 - 3tan²x}  = 3

tanx + 8tanx/{1 - 3tan²x}  = 3

tanx(1 - 3tan²x) + 8tanx/(1 - 3tan²x) = 3

9tanx - 3tan³x/(1 - 3tan²x) = 3

3(3tanx - tan³x) / 1 - 3tan²x = 3

[3tanx -  tan³x] / [1 - 3tan²x] = 1.

Hence proved.