Picture proof of trigonometric angle-sum formulas
You can drag points B, C, and D. Scroll down for more explanation. Based on an image by Tamás Görbe (twitter @TamasGorbe)
How to construct (and why it works):
1. Start with the bottom right triangle (ABC)
2. Draw the right triangle ACD with one leg on the hypoteneuse of ABC.
3. Construct the line through D parallel to AB.
4. Continue segment BC to meet that parallel line at E. BE is perpendicular to to DE by vertical angles, and angle DCE is equal to α because it is the complement of the complement of α.
5. Choose point F so that the segment AF is perpendicular to the line through D and E (from step 3). Angle ADF is equal to α+β by vertical angles.6. Fixing the length of AD at 1, the rest follows by basic trigonometry.