Trigonometric Functions — Maths STD 12 Science — Question

cos x - sin x = 1
Dividing by $\sqrt{1^2+(-1)}=\sqrt{2}$
$\frac{1}{\sqrt{2}} \cos x-\frac{1}{\sqrt{2}} \sin x=\frac{1}{\sqrt{2}}$
$\cos \frac{\pi}{4} \cos x -\sin \frac{\pi}{4} \sin x =\frac{1}{\sqrt{2}}$
$\cos \left( x +\frac{\pi}{4}\right)=\cos \frac{\pi}{4} \quad \ldots$ (i)
The general solution of $\cos \theta=\cos \alpha$ is $\theta=2 n \pi \pm \frac{\pi}{4} ; n \in z$
∴ The genera; solution of equation (i) given by 
$x +\frac{\pi}{4}=2 n \pi \pm \frac{\pi}{4} ; n \in z$
$x =2 n \pi ; x=2 n \pi-\frac{\pi}{2} ; n \in z$