Question 1011295
1. {{{cot^2 (theta )(csc theta)}}} in terms of {{{sin( theta )}}}

since
{{{cot^2 (theta) =cos^2(theta)/sin^2(theta)}}} and
{{{csc (theta)=1/sin(theta)}}}, we have


{{{cot^2 (theta) (csc (theta))=(cos^2(theta)/sin^2(theta))(1/sin(theta))}}}


since {{{cos^2(theta)=1-sin^2(theta)}}}, we have


{{{cot^2 (theta) (csc (theta))=((1-sin^2(theta))/sin^2(theta))(1/sin(theta))}}}


{{{cot^2 (theta) (csc (theta))=(1/sin^2(theta)-sin^2(theta)/sin^2(theta))(1/sin(theta))}}}

{{{cot^2 (theta) (csc (theta))=((1/sin^2(theta))-1)(1/sin(theta))}}}



2. 
if you have

{{{(1-csc^2 (theta ))/ csc^2  (theta )}}} , in terms of {{{cos (theta )}}} will be:

since {{{1/ csc^2  (theta )=sin^2(theta)}}}, we will have

{{{(1-csc^2 (theta ))(1/ csc^2  (theta )) =(1-csc^2 (theta ))sin^2(theta) }}}


since {{{csc^2 (theta )=1/sin^2(theta)}}}, we will have


{{{(1-csc^2 (theta ))(1/ csc^2  (theta )) =(1-1/sin^2(theta))sin^2(theta)}}}
 

{{{(1-csc^2 (theta ))(1/ csc^2  (theta )) =sin^2(theta)-sin^2(theta)/sin^2(theta) }}}


{{{(1-csc^2 (theta ))(1/ csc^2  (theta )) =sin^2(theta)-1}}}


{{{(1-csc^2 (theta ))(1/ csc^2  (theta )) =-(1-sin^2(theta))}}}


{{{(1-csc^2 (theta ))/ csc^2  (theta ) =-cos^2(theta)}}}