We know that,
#(1)tan2x=(2tanx)/(1-tan^2x)#
#(2)sin2x=(2tanx)/(1+tan^2x)#
Using #(1)and(2)#
#cot2theta+tantheta=1/tan(2theta)+tantheta#
#=1/((2tantheta)/(1-tan^2theta))+tantheta#
#=(1-tan^2theta)/(2tantheta)+tantheta#
#=(1-tan^2theta+2tan^2theta)/(2tantheta)#
#=(1+tan^2theta)/(2tantheta)#
#=1/((2tantheta)/(1+tan^2theta))#
#=1/(sin2theta)#
#=csc2theta#