We are given the equation:

$\sin(x) - 1 = \cos(x)$

Step 1: Rearrange the equation

First, isolate one trigonometric function on one side of the equation:

$\sin(x) = \cos(x) + 1$

Step 2: Use the Pythagorean identity

We know the Pythagorean identity:

$\sin^2(x) + \cos^2(x) = 1$

Now, square both sides of the equation $\sin(x) = \cos(x) + 1$:

$\sin^2(x) = (\cos(x) + 1)^2$

Step 3: Expand and simplify

Expand the right-hand side:

$\sin^2(x) = \cos^2(x) + 2\cos(x) + 1$

Now, substitute $\cos^2(x) = 1 - \sin^2(x)$ into the equation:

$\sin^2(x) = (1 - \sin^2(x)) + 2\cos(x) + 1$

Simplifying:

$\sin^2(x) = 2 - \sin^2(x) + 2\cos(x)$

Now, bring all terms involving $\sin(x)$ and $\cos(x)$ to one side:

$2\sin^2(x) - 2\cos(x) = 2$

Divide the whole equation by 2:

$\sin^2(x) - \cos(x) = 1$

Step 4: Solve using a numerical or graphical method

This is a non-linear trigonometric equation, and solving it algebraically is quite complex. To find the exact solutions, we would typically use numerical methods or graphing.

Would you like me to help you solve this equation using a graphing or numerical method?