Question

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

Answer 1

Please see below.

Explanation:

We know that,

`color(red)((1)a^3-b^3=(a-b)(a^2+ab+b^2)`

`color(blue)((2)csc^2x=1+cot^2x`

`color(violet)((3)tanxcotx=1`

Here,

`(tan^3x-cot^3x)/(tan^2x+csc^2x)=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`