I $intuit$ that the sum of accute angles of $2$ dissimilar Pythagorean triples cannot be a right angle because the sine and cosine of every triple is unique. I wrote a program to see if some combination of $3$ triples would work and I seem to have found many but I wanted to verify the results manually and here is the problem I found with $(3,4,5), (12,5,26), (63,16,65)$ where
$$\sin\alpha =\frac{4}{5}, \cos\alpha =\frac{3}{5}\quad \sin\beta=\frac{5}{13}, \cos\beta=\frac{12}{13}\quad \sin\gamma=\frac{16}{65}, \cos\gamma=\frac{63}{65}$$ I've been given to understand that $$\begin{align} \sin(\alpha+\beta+\gamma) &= \sin\alpha\cos\beta\cos\gamma + \cos\alpha\sin\beta\cos\gamma +\cos\alpha\cos\beta\sin\gamma - \sin\alpha\sin\beta\sin\gamma \end{align}$$ Then I worked out the following in Word/Excel
I pasted the first $2$ $equals$ into excel and got $1$ for each. Where did I go wrong in the last one?