Question

If tan x + tan 2x + tan 3x = 0, then x =?
$
  {\text{A}}{\text{. n}}\pi {\text{ + }}\dfrac{\pi }{3} \\
  {\text{B}}{\text{. n}}\pi + \dfrac{\pi }{4} \\
  {\text{C}}{\text{. n}}\pi {\text{ or n}}\pi \pm \dfrac{\pi }{3} \\
  {\text{D}}{\text{. 2n}}\pi \\
$

Answer 1

Hint: To find the value of x, we rearrange the given equation and convert it in terms of sin and cos functions. Then we apply the formulae of sin (a + b) and cos (a + b) and simplify.

Complete step-by-step answer:
Given data, tan x + tan 2x + tan 3x = 0
⟹tan x + tan 2x = -tan 3x
$ \Rightarrow \dfrac{{{\text{sin x}}}}{{{\text{cos x}}}} + \dfrac{{{\text{sin 2x}}}}{{{\text{cos 2x}}}} = - \dfrac{{{\text{sin 3x}}}}{{{\text{cos 3x}}}}$
$ \Rightarrow \dfrac{{{\text{sin x cos 2x + cos x sin 2x}}}}{{{\text{cos x cos 2x}}}} = - \dfrac{{{\text{sin 3x}}}}{{{\text{cos 3x}}}}$
We know the function Sin (a + b) is given as (Sin a Cos b + Cos a Sin b).
Comparing with the above equation, here a = x and b = 2x.
$ \Rightarrow {\text{sin }}\left( {{\text{2x + x}}} \right){\text{ cos 3x = - cos x cos 2x sin 3x}}$
$ \Rightarrow {\text{sin 3x cos 3x + cos x cos 2x sin 3x = 0}}$
$ \Rightarrow {\text{sin 3x }}\left( {{\text{cos 3x + cos x cos 2x}}} \right){\text{ = 0}}$
$ \Rightarrow {\text{sin 3x }}\left( {{\text{cos }}\left( {{\text{2x + x}}} \right){\text{ + cos x cos 2x}}} \right){\text{ = 0}}$
We know the function Cos (a + b) is given as (Cos a Cos b - Sin a Sin b).
Comparing with the above equation, here a = x and b = 2x
$ \Rightarrow {\text{sin 3x }}\left( {{\text{cos x cos 2x - sin x sin 2x + cos x cos 2x}}} \right) = {\text{ 0}}$
$ \Rightarrow {\text{ - sin 3x sin x sin 2x = 0}}$
$ \Rightarrow {\text{sin x sin 2x sin 3x = 0}}$

This means either of the terms is zero.
Either, sin 3x = 0
i.e., 3x = nπ, n ε I
i.e., x =$\dfrac{{{\text{n}}\pi }}{3}$, n ε I

Or, sin 2x = 0
i.e., 2x = nπ, n ε I
i.e., x =$\dfrac{{{\text{n}}\pi }}{2}$, n ε I
Or, sin x = 0
i.e., nπ, n ε I
But putting x =$\dfrac{{{\text{n}}\pi }}{2}$, does not satisfy the equation. (E.g. Putting x =$\dfrac{{{\text{n}}\pi }}{2}$, tan$\dfrac{\pi }{2}$= undefined)
Hence the solution set is x = nπ and x =$\dfrac{{{\text{n}}\pi }}{3}$
Hence Option C is the correct answer.

Note: In order to solve this type of problems the key is to have adequate knowledge in trigonometric formulae such as Sin (a + b) and Cos (a – b) and tan θ =$\dfrac{{{\text{Sin }}\theta }}{{{\text{Cos }}\theta }}$. It is important to identify that if${\text{sin x sin 2x sin 3x = 0}}$, then either of them is zero. Also Sin π = 0. And the letter I represents the set of integers here.