Mathematics - Trigonometric Equation Question with Solution | TestHub

Given, $\sin 2\mathrm{x}+2\sin \mathrm{x}+2\cos \mathrm{x}+1=0$ $\Rightarrow 1+\sin 2\mathrm{x}+2(\sin \mathrm{x}+\cos \mathrm{x})=0$ $\Rightarrow (\sin x+\cos x{)}^2+2(\sin x+\cos x)=0$ $\Rightarrow (\sin x+\cos x)(\sin x+\cos x+2)=0$ $\therefore \sin x+\cos x=0$ or $\sin x+\cos x=-2$ But, $\sin x+\cos x=-2$ is inadmissible. Since, $|\sin x|\le 1,|\cos x|\le 1$ $\therefore \sin x+\cos x=0\Rightarrow \sin \left(x+\frac{\pi }{4}\right)=0$ $\Rightarrow x+\frac{\pi }{4}=n\pi \Rightarrow x=n\pi -\frac{\pi }{4}$