Let AB is the diameter of the circle. Now draw two tangent PQ and RS at point A and B respectively.

The radii of the circle are OA and OB.

Now form figure,

OA is perpendicular to PQ and OB is perpendicular to RS.

Now ∠OBR = ∠OBS = ∠OAP = ∠OAQ = 90

Again from fiure, 

       ∠OBR = ∠OAQ    (Alternate interior angle)

and ∠OBR = ∠OAP    (Alternate interior angle)

Since alternate interior angles are equal. So Lines PQ and RS are parallel.

So the tangents drawn at the ends of a diameter of a circle are parallel.