\(\frac{cos θ [ ( 1 - cot θ) cot θ + cosec^2 θ]}{(1 - cos θ) cot θ (1 + cot θ) ( cosec θ + cot θ)}\) is equal to?

1

\(\frac{cos θ [ (cot θ - cot^2 θ + cosec^2 θ)}{( 1 - cos θ) cot θ (1 + cot θ) (cosec θ + cot θ)}\)

  (cosec² θ - cot² θ = 1, Multiplying an dividing the equation with cosec θ)

= \(\frac{cosec θ × cos θ (1+ cot θ)}{cosec θ ( 1 - cos θ) cot θ ( 1 + cot θ) ( cosec θ + cot θ)}\)

= \(\frac{cosec θ cos θ}{(cosec θ - cot θ) cot θ( cosec θ + cot θ)}\)

As, (a+ b)(a - b) = a² - b², cosec² θ - cot² θ = 1

= \(\frac{1}{sin θ}\) × \(\frac{cos θ}{cot θ}\) = \(\frac{cot  θ}{cot θ}\) = 1