Question: Show that
$$\arctan2 = \arccos(1/\sqrt5).$$
This is the result of solving one trigonometrical equation.

Answer: 
What you did is correct. Now
$$P(\mbox{alive} | \mbox{no water}) = 1- P(\mbox{die} | \mbox{no water}) = 1 - 0.9 = 0.1$$
and you can conclude the rest.
To produce the probability of plant being alive, use the numbers you have. 

(Certainty of Being Watered)(Plant will survive with Water)+(Certainty of Not being Watered)(Plant surviving without Water)