find all solutions of the equation in the interval 0, 2π). (enter your answers as a comma - separated list…

Explanation:

Step1: Rearrange the equation

We start with $\sin x - 1=\cos x - 1$. Add 1 to both sides to get $\sin x=\cos x$.

Step2: Use the tangent - relation

Since $\sin x=\cos x$ and $\cos x\neq0$ (if $\cos x = 0$, $\sin x=\pm1$ and the equation won't hold), we can divide both sides by $\cos x$. So, $\tan x = 1$.

Step3: Find solutions in the given interval

The tangent function $y = \tan x$ has a period of $\pi$. The solutions of $\tan x=1$ in the interval $[0,2\pi)$ are $x=\frac{\pi}{4}$ and $x=\frac{5\pi}{4}$.

Answer:

$\frac{\pi}{4},\frac{5\pi}{4}$