How to prove (tan^3x-cot^3x)/(tan^2x+csc^2x)=tanx-cotx?

1 Answer
Apr 29, 2018

Please see below.

Explanation:

We know that,

color(red)((1)a^3-b^3=(a-b)(a^2+ab+b^2)(1)a3b3=(ab)(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-cotxtan3xcot3xtan2x+csc2x=tanxcotx

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