Alright, lets get started.
Please refer the diagram, I have attached.
∠ABC is given as y.
So, the ∠EBC will be 180 - ∠ABC.
∠EBC = 180 - y
As given in question, BO is the bisector of ∠EBC, it means ∠OBC will be half of the ∠EBC.
∠OBC =
of ∠EBC
∠OBC =
(180 - y)
∠OBC = 
Similarly,
the ∠DCB will be 180 - ∠ACB
∠DCB = 180 - z
As given in question, CO is the bisector of ∠DCB, it means ∠OCB will be half of the ∠DCB.
∠OCB =
of ∠DCB
∠OCB =
(180 - z)
∠OCB = 
In triangle OBC, the sum of the angles of the triangle will be 180.
∠BOC +
+
= 180
∠BOC + 
∠BOC =
= 
As per upper triangle ABC, 
or 
Putting this value in value of ∠BOC
∠BOC = 
∠BOC = 
∠BOC = 90 -
: Hence proved
Hope it will help :)