solution: From the angle addition and subtraction formulas,
  \begin{align*}
  \sin (x + y) &= \sin x \cos y + \cos x \sin y, \\
  \sin (x - y) &= \sin x \cos y - \cos x \sin y,
  \end{align*}so
  \begin{align*}
  \sin (x + y) \sin (x - y) &= (\sin x \cos y + \cos x \sin y)(\sin x \cos y - \cos x \sin y) \\
  &= \sin^2 x \cos^2 y + \sin x \cos x \sin y \cos y - \sin x \cos x \sin y \cos y - \cos^2 x \sin^2 y \\
  &= \sin^2 x (1 - \sin^2 y) - (1 - \sin^2 x) \sin^2 y \\
  &= \sin^2 x - \sin^2 x \sin^2 y - \sin^2 y + \sin^2 x \sin^2 y \\
  &= \sin^2 x - \sin^2 y.
  \end{align*}Taking $x = \arcsin 0.5$ and $y = \arcsin 0.4,$ we get
  \begin{align*}
  \sin (\arcsin 0.5 + \arcsin 0.4) \cdot \sin (\arcsin 0.5 - \arcsin 0.4) &= \sin^2 (\arcsin 0.5) - \sin^2 (\arcsin 0.4) \\
  &= 0.5^2 - 0.4^2 \\
  &= 0.09 = \boxed{\frac{9}{100}}.
  \end{align*}
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Evaluate
  \[\sin (\arcsin 0.4 + \arcsin 0.5) \cdot \sin (\arcsin 0.5 - \arcsin 0.4).\]
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: \frac{9}{100}
extra_info.question: Evaluate
  \[\sin (\arcsin 0.4 + \arcsin 0.5) \cdot \sin (\arcsin 0.5 - \arcsin 0.4).\]
extra_info.split: test