Question

$4\sin x\cos x+2\sin x+2\cos x+1=0$.

Answer 1

$4\sin x\cos x+2\sin x+2\cos x+1=0\Rightarrow 2(\sin x+\cos x)+2\sin x\cos x+1+2\sin x\cos x+1-1=0\Rightarrow 2(\sin x+\cos x)+(2\sin x\cos x+\cos {}^{2}x+\sin {}^{2}x).2-1=0\Rightarrow 2(\sin x+\cos x)+2{(\cos x+\sin x)}^2-1=0$

Let $\cos x+\sin x=k$

$\Rightarrow 2k^2+2k-1=0\Rightarrow k=\frac{-2\pm \sqrt{12}}{2}=\frac{-1\pm \sqrt{3}}{2}\Rightarrow k=\frac{\sqrt{3}-1}{2}or\frac{-\sqrt{3}-1}{2}$

$\cos x+\sin x=\frac{\sqrt{3}-1}{2}or\cos x+\sin x=\frac{-\sqrt{3}-1}{2}for\cos x+\sin x=\frac{\sqrt{3}-1}{2},x=\frac{\pi }{2}+\frac{\pi }{6}=\frac{2\pi }{3}for\cos x+\sin x=\frac{-\sqrt{3}-1}{2},x=\frac{3\pi }{2}-\frac{\pi }{6}=\frac{10\pi }{6}$