How does WolframAlpha 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.

Call

$$ u = \cos^{-1}\frac{15}{17} $$

Therefore

$$ \cos u = \frac{15}{17} $$

and

$$ \sin u = \sqrt{1 - \cos^2 u} = \sqrt{1 - \frac{15^2}{17^2}} = \frac{8}{17} $$

With these two you just need to calculate

$$ \sin 2u = 2\sin u \cos u = 2\frac{15}{17}\frac{8}{17} = \color{blue}{\frac{240}{289}} $$

\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}