Question

How do you solve $\tan 4x=\tan 2x$?

Answer 1

Hint: In the given question, we are given an equation involving tangent trigonometric function with two different angles x and y which we need to equate. Now, we need to use the fact that the value of the tangent function repeats after one cycle from 0 to $2\pi $ and also make use of the domain of the tangent trigonometric function.

Complete step-by-step solution:
In the given question, we are given a trigonometric equation such that it involves a tangent function having two different angles 4x and 2x. Now, what we need to do is that we need to find all those angles such that it satisfies the equation $\tan 4x=\tan 2x$.
Now, we are having the equation $\tan 4x=\tan 2x$ and solving for this we proceed as follows:
$\begin{align}
  & \tan 4x=\tan 2x \\
 & \Rightarrow 4x=n\pi +2x \\
\end{align}$
Now, simplifying this by taking 3x to the left-hand side and then subtracting it from 4x we get $\Rightarrow 2x=n\pi $and now from this solving for x we get
$\Rightarrow x=\dfrac{n\pi }{2}$
Here wherever n is used everywhere n is taken from the set of integers, that is $n\in Z$.

Note: In the given question, we are given trigonometric equation so while solving this we require the domain and co-domain definition in order to define a generalised form of the domain of every angle and then in order to check the values and then equate it to another tangent function wisely. Also, we need to make sure that the angle chosen satisfies the equation.