Question

tanx+tan(x+pi/3)+tan(x+2pi/3)

Answer 1

HELLO DEAR,

YOUR QUESTIONS IS--------------If tan x + tan (x + π/3) + tan (x + 2π/3) = 3

prove that tan 3x = 1

we know:- tan(A + B) = [{tanA + tanB}/{1 - tanA.tanB}]

therefore,

tanx + {tanx + tanπ/3}/{1 - tanx.tanπ/3} + {tanx + tan2π/3}/{1 - tanx.tan2π/3} = 3

=> tanx + {tanx + √3}/{1 - √3tanx} + {tanx - √3}/{1 + √3tanx} = 3

=> $\bold{tanx + \frac{(tanx + \sqrt{3})(1 + \sqrt{3}tanx) + (tanx - \sqrt{3})(1 - \sqrt{3}tanx)}{(1 - \sqrt{3}tanx)(1 + \sqrt{3} tanx)}}$ = 3

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

=> tanx + {8tanx}/{1 - 3tan²x} = 3

=> {tanx - 3tan³x + 8tanx}/{1 - 3tan²x} = 3

=> {9tanx - 3tan³x}/{1 - 3tan²x} = 3

=> 3{tanx - 3tan³x}/{1 - 3tan²x} = 3

we know:- tan3A = {tanA - 3tan³A}/{1 - 3tan²A}

therefore, tan3x = 3/3

=> tan3x = 1 [hence, proved]

I HOPE ITS HELP YOU DEAR,

THANKS

Answer 2

please like my answer

hope this helps u