The value of $(\frac{1-cotθ}{1-tanθ})^2 +1$, if 0° < θ < 90°, is equal to: 

$cosec^2θ$

( \(\frac{1 - cotθ }{1 + tanθ}\) )² + 1

= ( \(\frac{1 - cosθ/sinθ }{1 + sinθ/cosθ}\) )² + 1 

= ( \(\frac{(sinθ - cosθ)² }{sin²θ}\)  ×  ( \(\frac{cos²θ }{ ( cosθ + sinθ )²}\) + 1

= ( \(\frac{cos² θ}{sin²θ}\) + 1

= cot²θ + 1

{ Using , cosec²θ - cot²θ = 1 }

= cosec²θ