**Solution:**

Let's consider a circle with center O and radius r. Two tangents are drawn from a point P outside the circle, touching the circle at points A and B. The tangents intersect at point T.

**1. Drawing the tangents:**
To draw the tangents, we can use the property that the tangent to a circle is perpendicular to the radius drawn to the point of contact.

- Draw the radius OP from the center of the circle to the point P outside the circle.
- At point P, draw a line perpendicular to OP.
- Mark the points of contact with the circle as A and B.

**2. Finding the angle between the tangents:**
Given that the angle between the tangents is 60 degrees, it means that angle ATB is 60 degrees.

**3. Connecting the points of contact:**
- Draw the radii OA and OB from the center O to the points of contact A and B, respectively.
- We need to find the angle between these radii.

**4. Finding the angle between the radii:**
- Let's denote the angle between the radii as x.
- Since OA and OB are radii, they are equal in length (OA = OB = r).
- Triangle OAB is an isosceles triangle since OA = OB.
- Therefore, angle OAB = angle OBA = (180 - x)/2 = 90 - x/2.

**5. Using the angle sum property:**
- In triangle OAB, angle OAB + angle OBA + angle AOB = 180 degrees.
- Substituting the values, we have (90 - x/2) + (90 - x/2) + 60 = 180.
- Simplifying the equation, we get 180 - x + 60 = 180.
- Combining like terms, we have 240 - x = 180.
- Solving for x, we get x = 240 - 180 = 60 degrees.

**6. Conclusion:**
Therefore, the angle between the radii OA and OB drawn from the points of contact A and B is 60 degrees.