Proof of angle addition formula for cosine  Trigonometry  Khan Academy  Summary and Q&A
TL;DR
This video provides a stepbystep proof of the angle addition formula for cosine, demonstrating that cosine of X plus Y is equal to cosine X times cosine Y minus sine X times sine Y.
Questions & Answers
Q: How is cosine of X plus Y expressed in terms of segments in the video?
Cosine of X plus Y is represented as the length of segment AF, which is equivalent to segment AB minus segment FB.
Q: How does the proof establish that segment AB is equal to cosine X times cosine Y?
By considering right triangle ACB, it is shown that segment AB, adjacent to angle Y, is equal to cosine of X times cosine of Y.
Q: How is segment FB defined in the proof?
Since ECBF is a rectangle, segment FB is equivalent to segment EC. By applying sine of Y, it is proven that segment FB is equal to sine X times sine Y.
Q: What is the final expression for cosine of X plus Y in the proof?
Cosine of X plus Y is equal to cosine X times cosine Y minus sine X times sine Y.
Summary & Key Takeaways

The video aims to prove the angle addition formula for cosine.

It uses triangles and trigonometric ratios to express cosine of X plus Y as the difference of two segments.

The proof shows that segment AB is equal to cosine X times cosine Y, and segment FB is equal to sine X times sine Y.

By subtracting FB from AB, the video concludes that cosine of X plus Y is equal to cosine X times cosine Y minus sine X times sine Y.