[tex]\dfrac{\csc^2\theta-3\csc\theta+2}{\csc^2\theta-1}[/tex]
Identity:
[tex]\sin^2\theta+\cos^2\theta=1\implies1+\cot^2\theta=\csc^2\theta[/tex]
So we can rewrite the denominator to get
[tex]\dfrac{\csc^2\theta-3\csc\theta+2}{\cot^2\theta}[/tex]
Multiply numerator and denominator by [tex]\sin^2\theta[/tex]. Several terms will cancel since [tex]\sin\theta\csc\theta=1[/tex]. Also, [tex]\cot\theta=\dfrac{\cos\theta}{\sin\theta}[/tex]. We get
[tex]\dfrac{1-3\sin\theta+2\sin^2\theta}{\cos^2\theta}[/tex]
Factorize the numerator, and write [tex]\cos[/tex] in terms of [tex]\sin[/tex] in the denominator to factorize it further to get
[tex]\dfrac{(1-\sin\theta)(1-2\sin\theta)}{\cos^2\theta}=\dfrac{(1-\sin\theta)(1-2\sin\theta)}{1-\sin^2\theta}=\dfrac{(1-\sin\theta)(1-2\sin\theta)}{(1-\sin\theta)(1+\sin\theta)}[/tex]
The [tex]1-\sin\theta[/tex] factors cancel, leaving you with
[tex]\dfrac{1-2\sin\theta}{1+\sin\theta}[/tex]
which you could simplify a bit further by writing
[tex]\dfrac{1+\sin\theta-3\sin\theta}{1+\sin\theta}=1-\dfrac{3\sin\theta}{1+\sin\theta}[/tex]
