HMMT 2015 U8

Evaluate

$$\sin (\arcsin (0.4)+\arcsin (0.5)) \cdot \sin (\arcsin (0.5)-\arcsin (0.4))$$

where for $x \in[-1,1], \arcsin (x)$ denotes the unique real number $y \in[-\pi, \pi]$ such that $\sin (y)=x$.