To find the angle between the pair of tangents, we first need to find the points where the tangents intersect the ellipse.

The equation of the ellipse is given by:
3x^2 + 2y^2 = 5

We can rewrite this equation as:
x^2/ (5/3) + y^2/ (5/2) = 1

Comparing this equation with the standard form of an ellipse:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1

We can see that the center of the ellipse is at (0,0) and the semi-major axis length (a) is sqrt(5/3) and the semi-minor axis length (b) is sqrt(5/2).

Now, let's find the points where the tangents intersect the ellipse.

The equation of a line passing through the point (1,2) is given by:
(y-2) = m(x-1)

where m is the slope of the line.

We can rewrite this equation as:
y = mx + (2-m)

Substituting this equation into the equation of the ellipse, we get:
3x^2 + 2(mx + (2-m))^2 = 5

Expanding and simplifying, we get:
3x^2 + 2(m^2x^2 + 4mx(2-m) + (2-m)^2) = 5
3x^2 + 2m^2x^2 + 8mx(2-m) + 2(2-m)^2 = 5
(3 + 2m^2)x^2 + 8mx(2-m) + 2(2-m)^2 - 5 = 0

This is a quadratic equation in x. For the tangents to intersect the ellipse, this quadratic equation must have equal roots.

Therefore, the discriminant of this quadratic equation must be zero:
(8m(2-m))^2 - 4(3 + 2m^2)(2(2-m)^2 - 5) = 0

Simplifying and solving for m, we get:
64m^2(2-m)^2 - 4(3 + 2m^2)(4 - 8m + 4m^2 - 5) = 0
64m^2(2-m)^2 - 4(3 + 2m^2)(-1 + 8m - 4m^2) = 0
64m^2(2-m)^2 + 4(3 + 2m^2)(1 - 8m + 4m^2) = 0
64m^2(2-m)^2 + 4(3 + 2m^2 - 8m + 4m^2) = 0
64m^2(2-m)^2 + 4(3 + 6m^2 - 8m) = 0
64m^2(2-m)^2 + 4(3 - 2m + 6m^2) = 0
64m^2(2-m)^2 + 12 - 8m + 24m^2 = 0
64m^2(2-m)^2 + 36m^2