[sin - cos + 1]/[sin + cos - 1] = [sin + 1]/cosiff [sin - cos + 1]*cos = [sin + 1]*[sin + cos - 1]
iff sin*cos - cos^2 + cos = sin^2 + sin*cos - sin + sin + cos - 1

iff -cos^2 = sin^2 - 1

1 = sin^2 + cos^2, which is true,

Manipulate normally, noting:

1. cot x = cos x / sin x
2. cos² x + sin² x = 1 → sin²x = 1 - cos² x
3. a² - b² = (a + b)(a - b)
4. 1 = 1²
5. ab = ba
6. a/(bc) = a/b/c

(1 + cot x)² - 2 cot x = 1² + 2 cot x + cot² x - 2 cot x

= 1 + cot² x

= 1 + (cos x / sin x)²

= 1 + cos² x / sin² x

= 1 + cos² x / (1 - cos² x)

= ((1 - cos² x) + cos² x)/(1 - cos² x)

= 1/(1² - cos² x)

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

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

QED.