We know that,
color(red)((1)a^3-b^3=(a-b)(a^2+ab+b^2)(1)a3−b3=(a−b)(a2+ab+b2)
color(blue)((2)csc^2x=1+cot^2x(2)csc2x=1+cot2x
color(violet)((3)tanxcotx=1(3)tanxcotx=1
Here,
(tan^3x-cot^3x)/(tan^2x+csc^2x)=tanx-cotxtan3x−cot3xtan2x+csc2x=tanx−cotx
Let,
LHS=color(red)((tan^3x-cot^3x))/(tan^2x+color(blue)(csc^2x))...toApply(1)and(2)
=color(red)(((tanx-cotx)(tan^2x+tanxcotx+cot^2x)))/(tan^2x+color(blue)(1+cot^2x))
=((tanx-cotx)(tan^2x+color(violet)(1)+cot^2x))/((tan^2x+1+cot^2x))...toApply(3)
=tanx-cotx
=RHS