if sec x = root 2, find the value of (1+tan x+cosec x) / (1+cotx-cosecx)?

secx = (2)^1/2
Hence cosx = 1/(2)^1/2
So here x = 45 degree

So Put x = 45 in the expression, we have
$$\begin{gathered} \frac{1+\mathrm{tanx} +\mathrm{cosecx}}{1+\mathrm{cotx}-\mathrm{cosecx}} \\ = \frac{1+\tan 45 +\mathrm{cosec}45}{1+\cot 45-\mathrm{cosec}45} \\ =\frac{1+1+\sqrt{2}}{1+1-\sqrt{2}} \\ =\frac{2+\sqrt{2}}{2-\sqrt{2}} \\ =\frac{2+\sqrt{2}}{2-\sqrt{2}}\times \frac{2+\sqrt{2}}{2+\sqrt{2}} \\ =\frac{(2+\sqrt{2}{)}^2}{2} \left(\mathrm{ans}\right) \end{gathered}$$