[tex]\frac{1 - cot(x)}{tan(x) - 1} = \frac{1 - \frac{1}{tan(x)}}{\frac{1}{cot(x)} - 1} = \frac{\frac{tan(x)}{tan(x)} - \frac{1}{tan(x)}}{\frac{1}{cot(x)} - \frac{cot(x)}{cot(x)}} = \frac{\frac{tan(x) - 1}{tan(x)}}{\frac{1 - cot(x)}{cot(x)}} = \frac{tan(x) - 1}{tan(x)} \times \frac{cot(x)}{1 - cot(x)} = \frac{tan(x) - 1}{1} \times \frac{cot^{2}(x)}{1 - cot(x)} = \frac{cot(x) - cot^{2}(x)}{[tan(x) - 1][1 - cot(x)]} = \frac{cot(x)[1 - cot(x)]}{[tan(x) - 1][1 - cot(x)]} = \frac{cot(x)}{tan(x) - 1}[/tex]
