Question

tanx + tan2x + tan3x = 0 . Find General solution for x

Answer 1

 tan x + tan 2x + tan 3x = 0 
or, tan x + tan 2x = -tan 3x 
or, sin x/cos x + sin 2x/cos 2x = -sin 3x/cos 3x 
or, (sin x *cos 2x + cos x * sin 2x )/cos x * cos 2x = -sin 3x/cos 3x 
or, sin (2x + x )*cos 3x = - cos x * cos 2x * sin 3x 
or, sin 3x * cos 3x + cos x * cos 2x * sin 3x = 0 
or, sin 3x (cos 3x + cos x * cos 2x)= 0 
or, sin 3x [cos ( 2x + x ) + cos x * cos 2x)= 0 
or, sin 3x [ cos x * cos 2x - sin x * sin 2x+ cos x * cos 2x)= 0 
or, - sin 3x * sin x * sin 2x = 0 
or, sin x * sin 2x * sin 3x = 0 

either, sin 3x = 0 
ie, 3x = nπ , n ε I 
ie, x = (nπ/3) , n ε I 

or, sin 2x = 0 
ie, 2x = nπ , n ε I 
ie, x = (nπ/2) , n ε I 

or, sin x = 0 
ie, nπ , n ε I 

But putting x = nπ/2 , does not satisfy the equation 
(eg. Putting x = π/2, tan π/2 = undefined) 
& the solution set x = nπ is already included in the set x = nπ/3 
so the required solution is , x = nπ/3 

hence the general solutions of x are, x = (nπ/3) n ε I ie. n = ....-3,-2,-1,0,1,2,3....... 

Answer 2

Step-by-step explanation:

answer is nπ/3 and X=nπ+-Alpha