Question

Solve the equation $\sin x+\cos x=\sin 2x-1$.

Answer 1

If $\sin x+\cos x=t$ then on squaring
$1+\sin 2x=t^2$
$\therefore t=(t^2-1)-1$ or $t^2-t-2=0$
or $(t-2)(t+1)=0$ $\therefore t=2,-1$
$\cos x+\sin x=2$ or $-1$
or $\frac{1}{\sqrt{2}}\cos x+\frac{1}{\sqrt{2}}\sin x=\sqrt{2}$ or $-\frac{1}{\sqrt{2}}$
or $\cos (x-\frac{\pi }{4})=\sqrt{2}$ or $-\cos \frac{\pi }{4}$
The first option is rejected as $\sqrt{2}>1$
$\therefore \cos (x-\frac{\pi }{4})=\cos (\pi -\frac{\pi }{4})=\cos \frac{3\pi }{4}$
$\therefore x-\frac{\pi }{4}=2n\pi \pm \frac{3\pi }{4}$
$\therefore x=2n\pi +\pi$ or $2n\pi -\frac{\pi }{2}$.