drawing tangents to a circle when the size of the angle between them is given

Question

How can we draw two tangents to a circle if we want the size of the angle between them to be equal to a given angle?

Here we cannot use the old conventional way of drawing tangents from a point.

Answer 1

Hint. Let $ O $ be the centre of the circle, and $ T, U $ be the points of contact of the tangents with the circle. If the angle between the tangents is $ \alpha $, then $\widehat{TOU} = \pi - \alpha $.

Answer 2

Let $O$, $T$ and $U$ be the points introduced in @Bernard's answer.
You can draw two rays starting from $O$ that the angle between them is $\pi-\alpha$. The rays will cross the circle in $T$ and $U$ respectively. Now draw a line which has $T$ on it and is perpendicular to the corresponding ray and draw another similar one for $U$. The lines will be tangents to the circle and the angle between them will be $\alpha$.