How do you solve \[\sin 2x = \sin x\]?

Hint: In the above question we are given an equation that is \[\sin 2x = \sin x\] which is the trigonometric equation and we are asked how to solve it. For approaching such kinds of questions one should know about the basic identities of the trigonometry that is useful in the solving of the \[\sin 2x\]. The identity is as follows \[\sin 2x = 2\sin x\cos x\]. Using this identity in the above given equation and further solving and simplifying the equation and then solving for the\[x\]. We can get the desired solution. Let us see the implementation of the identity and the further solving for the in the complete step by step solution.

Complete step by step solution:
In this question we are given a trigonometric equation \[\sin 2x = \sin x\] and we are asked to solve it that can be done with the help of the trigonometric identities and the further solving. We are using here the identity for the \[\sin 2x\] is \[\sin 2x = 2\sin x\cos x\] so using this identity in place of \[\sin 2x\] in the given equation \[\sin 2x = \sin x\] it becomes as follows-
\[2\sin x\cos x = \sin x\]
Now taking the \[\sin x\] common the further equation becomes –
\[
  2\sin x\cos x - \sin x = 0 \\
  \sin x\left( {2\cos x - 1} \right) = 0 \\
 \]
Now the two conditions arises that are as follows-
\[\sin x = 0\] and \[\left( {2\cos x - 1} \right) = 0\]
Now solving for \[x\] to fetch the value of it from the two conditions aroused that is done by-
\[
  \sin x = 0 \\
  x = k\pi ,k \in z \\
 \]Here \[k \in z\] symbolizes \[x\] that belong to the integers
Now for the second condition that is \[\left( {2\cos x - 1} \right) = 0\]
\[
  \left( {2\cos x - 1} \right) = 0 \\
  \cos x = \dfrac{1}{2} \\
  x = \pm \dfrac{\pi }{3} + 2k\pi \\
  x = \dfrac{\pi }{3}\left( {6k \pm 1} \right),k \in z \\
 \] Here \[k \in z\] symbolizes \[x\] that belong to the integers

So the resultant values of \[x\] for the given equation are \[\sin 2x = \sin x\] equal to \[x = k\pi \] and \[x = \dfrac{\pi }{3}\left( {6k \pm 1} \right),k \in z\]

Note :
While approaching such kinds of questions one should know the basic trigonometric identities for the angles such \[2x,3x\] which would make the question solving very easily and carefully. Also one should know the values of different trigonometric terms at various angles like \[{30^ \circ },{45^ \circ },{60^ \circ },{90^ \circ },{0^ \circ }\] that proves to be vital in the solving of the questions.