If a=0 what is the number of solutions of equation x-a=square root(x^2-1).

First, we'll impose the constraints of existence of the
square root:

x^2 - 1
>=0

The expression x^2 - 1 is positive over the
ranges (- infinite , -1] U [1 ; +infinite).

Now, we'll put
a = 0 and we'll solve the given equation:

x = sqrt(x^2 -
1)

We'll raise to square both sides, in order to eliminate
the square root:

x^2 = x^2 -
1

We'll subtract x^2:

x^2 -
x^2 = -1

0 = -1 not
true

The given equation has no solutions if a
= 0.