Start with a circle of any radius. It can be centered at any point. Plot points A, B, C anywhere you like around the circle. Form triangle ABC.
The drawing isn't perfect and not to scale, but that doesn't matter. The drawing is a rough tool to help get the idea of what is going on. Form tangents at points B and C. Tangents are such that they meet the radius at right angles. Tangent lines only touch the circle at exactly one point.
Point D is the intersection of the tangent lines. Point E is the center of the circle. Angles EBD and ECD are 90 degrees each.
We are told that angle CAB is 50 degrees which is the angle shown in red. By the inscribed angle theorem we double that to get the measure of the blue arc along the circle from B to C. We follow the shortest path, aka minor arc. The inscribed angle CAB cuts off or subtends minor arc BC. The central angle BEC is equal to that of the measure of minor arc BC. So angle BEC is 100 degrees.
Focus on quadrilateral BECD in green.
We know three of the angles
angle BEC = 100 degrees
angle EBD = 90 degrees
angle ECD = 90 degrees
The unknown angle BDC = x is what we want to find
Recall that for any quadrilateral, the four angles always add to 360 degrees
(angle1)+(angle2)+(angle3)+(angle4) = 360
(angle BEC)+(angle EBD)+(angle ECD)+(angle BDC) = 360
100 + 90 + 90 + x = 360
280 + x = 360
280 + x - 280 = 360-280
x = 80
A shortcut is to realize that the 100 and x angles are supplementary, so
x+100 = 180
x+100-100 = 180-100
x = 80
We get the same x value
Answer: The angle between the tangent lines is 80 degrees.