Show that:cosec^2x-cosecx=cot^2x/1+

Question

Show that: cosec^2x-cosecx=cot^2x/1+sinx

Answer 1

csc^2(x) - csc(x) =

csc^2(x) (1 - 1/csc(x)) =

csc^2(x)(1 - sin(x)) =

csc^2(x)(1 - sin(x))(1 + sin(x))/(1 + sin(x)) =

csc^2(x)(1 - sin^2(x))/(1 + sin(x)) =

csc^2(x)cos^2(x)/(1 + sin(x)) =

(cos^2(x)/sin^2(x))/(1 + sin(x)) =

cot^2(x)/(1 + sin(x))

The work was guided by the fact that we knew that we wanted 1 + sin(x) in the denominator, and we were aware of the fact that (1 + sin(x))(1 - sin(x)) = 1 - sin^2(x) = cos^2(x) - hence, the factoring of the left side as we did.