Answer: The answer is (A) \(0,~\dfrac{\pi}{2}.\)
Step-by-step explanation: The given trigonometric equation is
\(\tan 2x\sin x=\tan 2x.\)
We are to solve the above equation in the interval \([0,2\pi).\)
The solution is as follows:
\(\tan 2x\sin x=\tan 2x\\\\\Rightarrow \tan2x\sin x-\tan 2x=0\\\\\Rightarrow \tan 2x(\sin x-1)=0\\\\\Rightarrow \tan 2x=0,~~~~~~~\sin x-1=0\\\\\Rightarrow \tan 2x=\tan 0,~\Rightarrow \sin x=1\\\\\Rightarrow 2x=0,~~~~~~~\Rightarrow \sin x=\sin \dfrac{\pi}{2}\\\\\\\Rightarrow x=0,~~~~~~~~~~\Rightarrow x=\dfrac{\pi}{2}.\)
Thus, the correct option is (A).
