Coincidental Trigonometric Identity for Two Particular Values I noticed that $$\sin(a+b)\sin(a-b) = \cos a \cos b\qquad (1)$$
when $$(a,b)=\left(\frac{2\pi}{5},\frac{\pi}{3}\right)$$
Is there an underlying reason for this coincidence?
Concretely, I would like to answer these questions:


*

*Does the solution set $(a,b)$ for $(1)$ permit a simple description?

*If $a$ is commensurable with $\pi$, does the same hold for $b$?


The left-hand side of $(1)$ equals $\cos^2 b−\cos^2 a$. Hence we have: $\cos b = x \cos a$, where $x^2−x−1=0$.
The answer to your second question looks like "no". At least Wolfram Alpha says nothing about the rationality of $\frac{1}{\pi}\arccos \frac{\sqrt{5}-1}{2}$.
