Question

Find the general solution of : $\cos x-\sin x=1$.

Answer 1

Given, $\cos x-\sin x=1$

Dividing both sides by $\frac{1}{\sqrt{2}}$

$\therefore \frac{1}{\sqrt{2}}\cos x-\frac{1}{\sqrt{2}}\sin x=\frac{1}{\sqrt{2}}$

$\therefore \cos (x+\frac{\pi }{4})=\cos \frac{\pi }{4}$

$x+\frac{\pi }{4}=2n\pi \pm \frac{\pi }{4}$, where $n\epsilon I$

$x=2n\pi \pm \frac{\pi }{4}-\frac{\pi }{4}$, where $n\epsilon I$

$x=2n\pi ,2n\pi -\frac{\pi }{4}-\frac{\pi }{4}$, where $n\epsilon I$

$x=2n\pi ,2n\pi -\frac{2\pi }{4}$, where $n\epsilon I$

$x=2n\pi ,2n\pi -\frac{\pi }{2}$, where $n\epsilon I$

$x=2n\pi ,\frac{\pi }{2}(4n-1)$, where $n\epsilon I$

Hence, the general solution is $2n\pi ,\frac{\pi }{2}(4n-1)$ where $n\epsilon I$.