Question

tan x/1- cotx+cot x/1- tan x= 1 + sec x cosec x.​

Answer 1

Answer

1 + sec x . cosec x

Explanation

$\rm LHS\\\\\implies \rm \dfrac{tan\ x}{1-cot\ x}+\dfrac{cot\ x}{1-tan\ x}\\\\\implies \rm \dfrac{tan^2x}{tan\ x-1}+\dfrac{cot\ x}{1-tan\ x}\ \;[LCM]\\\\\implies \rm \dfrac{cot\ x}{1-tan\ x}-\dfrac{tan^2x}{1-tan\ x}\ \; [Taking\ -ve\ sign\ common]\\\\\implies \rm \dfrac{1-tan^3x}{tanx(1-tan\ x)}\ \; [LCM]\\\\\implies \rm \dfrac{\cancel{(1-tan\ x)}(1+tan\ x+tan^2x)}{tan\ x(\cancel{1-tan\ x})}\ \; [a^3-b^3=(a-b)(a^2+ab+b^2)]\\\\\implies \rm \dfrac{tan\ x+sec^2x}{tan\ x}\ \; [sec^2x=1+tan^2x]$

$\implies \rm \dfrac{tan\ x}{tan\ x}+\dfrac{sec^2x}{tan\ x}\\\\\implies \rm 1+sec^2x\bigg( \dfrac{cos\ x}{sin\ x}\bigg)\\\\\implies \rm 1+sec\ x.\dfrac{1}{sin\ x}\\\\\implies \rm 1+sec\ x.csc\ x\\\\\implies \rm RHS$