Question

Solve the following equations.
$\frac{cot2x}{cotx}+\frac{cotx}{cot2x}+2=0.$

Answer 1

$\frac{cot2x}{cotx}+\frac{cotx}{cot2x}+2=0$

$\frac{co{t}^{2}2x+co{t}^{2}x+2cotxcot2x}{cotxcot2x}=0$

$\frac{{(cotx+cot2x)}^2}{cotx+cot2x}=0$

$cotx+cot2x=0$

$[cotx\ne 0\Rightarrow x\ne (2n+1)\pi /2$

$cot2x\ne 0\Rightarrow 2x\ne (2x+1)\pi /2x\ne (2n+1)\pi /2]$

$cotx+\frac{1}{2}(cotx-tanx)=0$

$3cotx-tanx=0$

$3-ta{n}^{2}x=0$

$[\therefore cotx=1/tanx]$

$tanx=\pm \sqrt{3}$

$x=(2n+1)\pi /3$