Actually many people missed a lot of FACTs. I came here late, but I have to do something to correct this issue.
FIRST, proving the Pythagorean Theorem and the Pythagorean Identity using trigonometry was done more than 150 years earlier. Loomis noted Versluys' book in his 1940 second edition, saying Versluys was not correct; however, Versluys was very correct (and of course Loomis was completely wrong) that at least in 1849 de Morgan published a book in which he showed that the angle sum identity for cos() without using the Pythagorean Theorem and the Pythagorean Identity. The angle sum identity was used in Versluys book. You can find Versluys' and de Morgan's 1849 books on the web easily.
SECOND, a high school girl, Rachel I. Mason, proved the Pythagorean Identity using trigonometry in 1996. However, she did not receive the same media coverage as what Jackson and Johnson did. In fact, in the late 1990s and the early 21th century, many people proved the Pythagorean identity using trigonometry. You can many such PUBLICATIONs on jstor.org; but you will need an account to download the articles.
THIRD, Jackson-Johnson's proof was not so great because it is long and complicated, compared with the proofs before their work. Actually, Zimba's 2009 work wasn't that creative either, because angle sum and angle difference identities were studied to death from the mid 1850s to the first half of the 20th century. More precisely, Zimba's theory only works on the interval (0,\pi/2) and he critized earlier work such as Versluys' based his a little bit restriction.
With the concept of limit and differentiation/integration, new techniques allow us to prove the Pythagorean Identity easily. SO, DON'T BE FOOLED BY THE MEDIA AND PEOPLE WHO DO NOT KNOW MUCH IN MATH HISTORY. Here is my newly created site, which is still under construction for various trigonometric proofs of the Pythagorean Identity: https://pages.mtu.edu/~shene/VIDEOS/GEOMETRY/GEO-HTML/Tri-Proofs/Trig-Proofs.html. A few people said this is a conspiracy-type-rambling. That made me LOL!
I do not create this site for fame. Instead, what I have been trying is to revealing the true and usually cold facts that many of us did not study enough and blindly following the media and some people who love media hype.
Please visit my site to see how the media did a dis-service to the math community.
