Question

2sin2x+sinx=4cosx+1.................................

Answer 1

$2\sin2x+\sin x=4\cos x+1\\ 4\sin x \cos x+\sin x=4\cos x+1\\ 4\sin x \cos x-4\cos x+\sin x-1=0\\ 4\cos x(\sin x-1)+1(\sin x-1)=0\\ (4\cos x +1)(\sin x-1)=0\\\\ 4\cos x+1=0\\ 4\cos x=-1\\ \cos x=-\dfrac{1}{4}\\ x=\cos^{-1}\left(-\dfrac{1}{4}\right)-2k\pi\\ x=\cos^{-1}\left(-\dfrac{1}{4}\right)+2k\pi\\\\ \sin x-1=0\\ \sin x=1\\ x=\dfrac{\pi}{2}+2k\pi\\\\ \boxed{x=\left\{\cos^{-1}\left(-\dfrac{1}{4}\right)-2k\pi,\cos^{-1}\left(-\dfrac{1}{4}\right)+2k\pi,\dfrac{\pi}{2}+2k\pi, (k\in \mathbb{Z})\right\}}$