How does WolframAlpha get an exact answer here? ${}{}$ 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?

Here are three relevant formulas:


*

*$\sin 2x = 2 \sin x \cos x$.

*$\cos \cos^{-1} x = x.$

*$\sin \cos^{-1} x = \sqrt{1 - x^2}$ after drawing an appropriate right triangle.
Combining these three to get the desired conclusion is left to the interested reader.
