Tangent of a Difference: Trig Identity Puzzle + Proof!

Carefully observe the diagram below. Notice . Slide the black slider slowly and CAREFULLY OBSERVE THE DYNAMICS THAT TAKE PLACE. After doing so, answer the questions and complete the activity prompts that appear below this applet.

Note: AE = 1.


What can we conclude about and ? Why can we conclude this?


Given your response to (1), write an equation that expresses the relationship among the lengths AF, AB, CD, and DB.


The 4 text boxes (containing trigonometric expressions) that appeared at the end of these dynamics represent the lengths of the 4 BOLD SEGMENTS you see in this diagram. Drag each expression next to its appropriate segment in this diagram.


Use your results from (3) and (2) to write an expression for in terms of these expressions (within the given diagram). What do you get?

Solution. (Don't watch until you've completed the exercises above!)