The `lim_(xto(pi)/2)(cot x-cosx)/((pi-2x)^(3))` equals

Answer

C

Explanation

`underset(xto(pi)/(2))lim(cotx-cosx)/((pi-2x)^(3))=underset(xto(pi)/(2))lim(cotx(1-sinx))/(8((pi)/(2)-x))`
`=(1)/(8)underset(xto(pi)/(2))lim(tan((pi)/(2)-x))/(((pi)/(2)-x)).((1-sin^(2)x))/(((pi)/(2)-x)^(2)).(1)/(1+sinx)`
`=(1)/(8)underset(xto(pi)/(2))lim(tan((pi)/(2)-x))/(((pi)/(2)-x)).(cos^(2)x)/(((pi)/(2)-x)^(2)).(1)/(1+sinx)`
`=(1)/(8)underset(xto(pi)/(2))lim(tan((pi)/(2)-x))/(((pi)/(2)-x)).(sin^(2)((pi)/(2)-x))/(((pi)/(2)-x)^(2)).(1)/(1+sinx)`
`=(1)/(8).1.1.(1)/(2)=(1)/(16)`