Question

The general solution of $ \dfrac{\tan 2x-\tan x}{1+\tan x\tan 2x}=1 $ is
 $ \begin{align}
  & \text{A}\text{. n}\pi \text{+}\dfrac{\pi }{4},\forall n\in Z \\
 & \text{B}\text{. n}\pi \pm \dfrac{\pi }{4},\forall n\in Z \\
 & \text{C}\text{. }\varnothing \\
 & \text{D}\text{. n}\pi \text{+}\dfrac{\pi }{6},\forall n\in Z \\
\end{align} $

Answer 1

Hint: To solve this question first of all we simplify the given equation by using the trigonometric formula of the tangent which is given as $ \tan \left( A-B \right)=\dfrac{\tan A-\tan B}{1+\tan A\tan B} $ . Then by using the trigonometric ratio table we will find the value and then we will find the general solution of the given equation.

Complete step by step answer:
We have been given an equation $ \dfrac{\tan 2x-\tan x}{1+\tan x\tan 2x}=1 $ .
We have to find the general solution of the given equation.
Now, let us consider the equation $ \dfrac{\tan 2x-\tan x}{1+\tan x\tan 2x}=1 $
Now, we know that in trigonometry the tangent formula is given as $ \tan \left( A-B \right)=\dfrac{\tan A-\tan B}{1+\tan A\tan B} $
Now, when we compare the formula with the given equation we can write the equation as
 $ \begin{align}
  & \Rightarrow \dfrac{\tan 2x-\tan x}{1+\tan 2x\tan x}=1 \\
 & \Rightarrow \tan \left( 2x-x \right)=1 \\
 & \Rightarrow \tan x=1 \\
\end{align} $
Now, we know that from trigonometric ratio table $ \tan \dfrac{\pi }{4}=1 $
Now, substituting the value in above equation we get
 $ \Rightarrow \tan x=\tan \dfrac{\pi }{4} $
Or we can write as $ x=\dfrac{\pi }{4} $
Now, we know that if $ \theta \And \alpha $ are not the multiples of $ \dfrac{\pi }{2} $ then we have
 $ \begin{align}
  & \Rightarrow \tan \theta =\tan \alpha \\
 & \Rightarrow \theta =n\pi +\alpha ,\forall n\in Z \\
\end{align} $
Where, Z is the set of integers.
Now, we can write the obtained equation as $ \Rightarrow x=n\pi +\dfrac{\pi }{4} $
So, the general solution of the above equation is $ \Rightarrow x=n\pi +\dfrac{\pi }{4},\forall n\in Z $
Option A is the correct answer.

Note:
 The expression of a trigonometric equations solution which involves the integer n is called the general solution of the equation. Be careful while solving, avoid common calculation mistakes. Also, remember trigonometric identities, trigonometric ratios and identities to solve these types of questions. Also try to remember some common general solutions of the equations to solve easily.