Converting polar equations to rectangular coordinates?

Question

Why does r = 1/ [sin(theta) - cos(theta)] become y - x = 1

Answer 1

r = 1 / [ sin θ - cos θ ]
x = r cos θ = cos θ / [ sin θ- cos θ ]
y = r sin θ = sin θ / [ sin θ- cos θ ]

y - x = ( sin θ- cos θ ) / (sin θ- cos θ )
y - x = 1

Answer 2

r = 1/ [sinθ - cosθ]
r[sinθ - cosθ] = 1
rsinθ - rcosθ = 1
but
rsinθ = y
and
rcosθ = x
so
y - x = 1