How do you verify the identity ##(csc x – cot x)^2 = (1 – cos x)/(1+cosx)##?

Start from the left side.
Since the right side contains cosines, rewrite the cosecant and cotangent functions as sines and cosines.
##(cscx-cotx)^2=(1/sinx-cosx/sinx)^2=(1-cosx)^2/sin^2x##
Use the identity ##sin^2x+cos^2x=1##.
##(1-cosx)^2/sin^2x=(1-cosx)^2/(1-cos^2x)##
Remember that ##a^2-b^2=(a-b)(a+b)##.
##(1-cosx)^2/(1-cos^2x)=(1-cosx)^2/((1-cosx)(1+cosx))=(1-cosx)/(1+cosx)##
Hence
##(cscx-cotx)^2=(1-cosx)/(1+cosx)##