I am a little confused with the problem sin^{2}(x/2) = (1-cos(x))/2 the half angle identity is used to get to (sqrt((1-cosx)/2))^{2}=(1-cos(x))/2). It appears to me that this can just be simplified to (1-cos(x))/2) = (1-cos(x))/2) because the exponent cancels the square root if I am wrong, why?

Something more complicated would be great explained step by step like cot(x/2) = (1+cos(x))/sinx as I get very stuck while trying to solve these.

thanks!