Solve the following equations.
$3^{logtanx}-{2.3}^{logcotx+1}=1$

$x=lo{g}_3^2$

$3^{logtanx}-{2.3}^{logcotx+1}=1$

$3^{logtanx}-{\mathrm{2.3.3}}^{log\frac{1}{tanx}}=1$

$3^{logtanx}-{6.3}^{-logtanx}=1$

Let $3^{logtanx}=t$

$t-\frac{6}{t}=1\Rightarrow t^2-t-6=0$

$t^2-3t+2t-6=0$

$t(t-3)+2(t-3)=0$

$(t-3)(t+2)=0$

$t=3,t=2$

$logtanx$

$3=3^1$

$logtanx=1$

$tanx=10$

$x=ta{n}^{-1}\left(10\right)$

$3^{logtanx}=2$

$logtanx=lo{g}^2$

$tanx={10}^{log{3}^2}$