Question

Solve $\sin x+\cos x=1+\sin x.\cos x$, then $2n\pi \pm \frac{\pi }{4}$
If true then enter $1$ and if false then enter $0$

Answer 1

$\because \sin x\cos x=1\sin x.\cos x$
Let $\sin x+\cos x=t\Rightarrow {\sin }^{2}x+{\cos }^{2}x+2\sin x.\cos x=t^2\Rightarrow \sin x.\cos s=\frac{t^2-1}{2}$
Now Put $\sin x+cosx=t$ and $\sin x.\cos x=\frac{t^2-1}{2}$ in (i), we get $t=1+\frac{t^2-1}{2}\Rightarrow t^{2}2t+1=0\Rightarrow t=1\Rightarrow \sin x\cos x=1\because t=\sin x+\cos x\Rightarrow \sin x+\cos x=1$
divide both sides of equation (ii) by $\sqrt{2},$ we get
$\Rightarrow \sin x\frac{1}{\sqrt{2}}+\cos x.\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}}\Rightarrow \cos (x-\frac{\pi }{4})=\cos \frac{\pi }{4}\Rightarrow x-\frac{\pi }{4}\Rightarrow x-\frac{\pi }{4}=2n\pi \pm \frac{\pi }{4}$