(1)/(sin(x)-cos(x))=cos(x)+sin(x)

What are we doing, solving for x?

then ....

sin^2 x - cos^2 x = 1
- cos 2x = 1
cos 2x = -1
2x = π
x = π/2 (90°)

since the period of cos 2x is π
other solutions are:
π/2 + π = 3π/2
3π/2 + π = 5π/2 , etc
x = ±π/2, ±3π/2, ±5π/2 , etc

in degrees, x = ±90°, ±270° , ±450° , etc