The value of 1/(cosecθ - cotθ) - 1/(sinθ) is:

\(\frac{1}{cosecθ - cotθ}\) - \(\frac{1}{sinθ}\) = \(\frac{sinθ}{1-cosθ}\) - \(\frac{1}{sinθ}\)

= \(\frac{sin^2θ - 1 + cosθ}{sinθ (1-cosθ)}\) = \(\frac{cosθ (1-cos^2θ)}{sinθ (1-cosθ)}\)

because sin²θ = 1-cos²θ

= \(\frac{cosθ}{sinθ}\) (\(\frac{1-cos}{1-cosθ}\)) = cotθ