I created a circle. Then clicked on the tangent (PcC) tool. Clicked to make the first point as prompted. When the next prompt appears in the command line I then click on the circle. It does not select nor does a line get created. What step am I missing?
Your procedure looks right. It should work.
LibreCAD sometimes has difficulties with tangents. Is the circle very small, very large or very far from absolute zero? Does it work on other circles, or in a fresh drawing? Is the drawing from a different source?
The tangent line should already appear when you have defined the starting point and then move the cursor towards the circle, you don't have to click on it exactly.
Try it with Free Snap on, otherwise snap might stick to somewhere not wanted and prevent proper working.
Another possibility: Active layer switched invisible and circle is on a different one.