Question

Solve the following expression and obtain the value of $x$
$\tan x+\tan 2x+\tan x\tan 2x=1$

Answer 1

Hint: We see that in the given expression $\tan x$ and $\tan 2x$ are present. Therefore, we can use the formula for tangent of sum of angles, i.e. $\tan \left( a+b \right)=\dfrac{\tan \left( a \right)+\tan \left( b \right)}{1-\tan \left( a \right)\tan \left( b \right)}$ and try to convert the given equation into that from to get the value of $\tan 3x$ from which we can get the value of x as the general solution for finding the angle for a given value of its tangent.

Complete step-by-step answer:
The given expression is
$\begin{aligned} & \tan \left( x \right)+\tan \left( 2x \right)+\tan \left( x \right)\tan \left( 2x \right)=1 \\\ & \Rightarrow \tan \left( x \right)+\tan \left( 2x \right)=1-\tan \left( x \right)\tan \left( 2x \right) \\\ & \Rightarrow \dfrac{\tan \left( x \right)+\tan \left( 2x \right)}{1-\tan \left( x \right)\tan \left( 2x \right)}=1......(1.1) \\\ \end{aligned}$
We know that the formula for $\tan \left( a+b \right)$ is given by
$\tan \left( a+b \right)=\dfrac{\tan \left( a \right)+\tan \left( b \right)}{1-\tan \left( a \right)\tan \left( b \right)}.............(1.2)$
Therefore, using $a=x$ and $b=2x$ in (1.2), we obtain
$\tan \left( 3x \right)=\tan \left( 2x+x \right)=\dfrac{\tan \left( x \right)+\tan \left( 2x \right)}{1-\tan \left( x \right)\tan \left( 2x \right)}$
Using this expression in equation (1.1), we can rewrite it as
$\tan \left( 3x \right)=1...............(1.3)$
However, we know that $\tan \left( \dfrac{\pi }{4} \right)=1$ and as the tan function has a periodicity of $\pi$, therefore, the tangent of two angles differing by a multiple of $\pi$ , i.e. $n\pi$ where n is an integer would be same. Thus, as the tangent of 3x and $\dfrac{\pi }{4}$ are the same, 3x and $\dfrac{\pi }{4}$ can differ only by a multiple of $\pi$ . Thus, we can write
$\begin{aligned} & \tan \left( 3x \right)=\tan \left( \dfrac{\pi }{4} \right) \\\ & \Rightarrow 3x=\dfrac{\pi }{4}+n\pi \\\ & \Rightarrow x=\dfrac{\pi }{12}+\dfrac{n\pi }{3} \\\ \end{aligned}$
Therefore, the answer to this question should be $x=\dfrac{\pi }{12}+\dfrac{n\pi }{3}$.

Note: In this question, we could also have found out the value of $\tan \left( x \right)$ by using the formula for $\tan \left( 2x \right)=\dfrac{2\tan \left( x \right)}{1-{{\tan }^{2}}\left( x \right)}$ to replace $\tan \left( 2x \right)$ by $\tan \left( x \right)$. We would then get a cubic equation, which we can solve for $\tan \left( x \right)$ and obtain the value of x from it. However, this method would be very lengthy and difficult to solve and will give the same answer as obtained in the solution. Therefore, in these types of questions, the method used in the solution is a faster and easier approach to get the value of x.