#(1-x^2)^(1/2)-x^2(1-x^2)^(-3/2)#
We will use : #color(red)(a^(-n) = 1/a^n)#
#<=> (1-x^2)^(1/2)-x^2/(1-x^2)^(color(red)(+3/2))#
We want two fractions with the same denominator.
#<=> ((1-x^2)^(1/2)*color(green)((1-x^2)^(3/2)))/color(green)((1-x^2)^(3/2))-x^2/(1-x^2)^(+3/2)#
We will use : #color(red)(u^(a)*u^(b) = u^(a+b))#
#<=> (color(red)((1-x^2)^(2)))/(1-x^2)^(3/2)-x^2/(1-x^2)^(3/2)#
#<=> ((1-x^2)^(2)-x^2)/(1-x^2)^(3/2)#
We will use the following polynomial identity :
#color(blue)((a+b)(a-b)=a^2-b^2)#
#<=> color(blue)((1-x^2+x)(1-x^2-x))/(1-x^2)^(3/2)#
#<=> ((-x^2+x+1)(-x^2-x+1))/(1-x^2)^(3/2)#
We can't do better than this, and now you can easily (if you want) find the solution of # ((-x^2+x+1)(-x^2-x+1))/(1-x^2)^(3/2) = 0#
