You found that when secants/chords meet <u>inside</u> the circle, the angle where they meet is <em>half the </em><u><em>sum</em></u> of the intercepted arcs.
When the secants/tangents meet <u>outside</u> the circle, the angle where they meet is <em>half the </em><u><em>difference</em></u> of the intercepted arcs.
The working shown is good as far as it goes. The difference needs to be multiplied by 1/2.