TITLE: How does WolframAlpha get an exact answer here? ${}{}$
QUESTION [29 upvotes]: I had a simple thing to compute with a calculator:
$$\sin\left(2\cos^{-1}\left(\frac{15}{17}\right)\right)$$
I got the decimal answer of about $0.83044983$, but when I typed it in WolframAlpha, it also gave an exact answer of $\frac{240}{289}$. How in the world would one get an exact answer here?

REPLY [7 votes]: \begin{align}
   \sin \theta &= \dfrac{8}{17} \\
   \cos \theta &= \dfrac{15}{17} \\
\hline
   \sin\left(2 \arccos \dfrac{15}{17} \right) 
   &= \sin(2 \theta) \\
   &= 2 \sin(\theta) \cos(\theta) \\
   &= \cdots
\end{align}