Alright, let's tackle this problem step by step. The question is to evaluate the expression:

\[\sin (\arcsin 0.4 + \arcsin 0.5) \cdot \sin (\arcsin 0.5 - \arcsin 0.4).\]

Hmm, okay. So, we need to compute the product of two sine terms, each of which involves the sum and difference of two arcsin functions. Let me recall some trigonometric identities that might help here. 

First, I remember that there's a formula for the sine of a sum and the sine of a difference. Specifically:

\[\sin(A + B) = \sin A \cos B + \cos A \sin B,\]
\[\sin(A - B) = \sin A \cos B - \cos A \sin B.\]

But in this case, the arguments inside the sine functions are themselves arcsin values. Let me denote:

Let’s set \(\alpha = \arcsin 0.4\) and \(\beta = \arcsin 0.5\). Then, the expression becomes:

\[\sin(\alpha + \beta) \cdot \sin(\beta - \alpha).\]

So, substituting these, we can compute each sine term separately and then multiply them. Let me compute each part step by step.

First, let's find \(\sin(\alpha + \beta)\). Using the sine addition formula:

\[\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta.\]

We know that \(\sin \alpha = 0.4\) and \(\sin \beta = 0.5\). But we need \(\cos \alpha\) and \(\cos \beta\). Since \(\alpha = \arcsin 0.4\), \(\cos \alpha\) can be found using the identity \(\cos^2 \theta + \sin^2 \theta = 1\). Therefore:

\[\cos \alpha = \sqrt{1 - \sin^2 \alpha} = \sqrt{1 - 0.4^2} = \sqrt{1 - 0.16} = \sqrt{0.84}.\]

Similarly,

\[\cos \beta = \sqrt{1 - \sin^2 \beta} = \sqrt{1 - 0.5^2} = \sqrt{1 - 0.25} = \sqrt{0.75}.\]

Calculating those square roots numerically:

\[\sqrt{0.84} \approx 0.9165,\]
\[\sqrt{0.75} \approx 0.8660.\]

So plugging back into the sine addition formula:

\[\sin(\alpha + \beta) = (0.4)(0.8660) + (0.9165)(0.5).\]

Calculating each term:

First term: \(0.4 * 0.8660 ≈ 0.3464\),
Second term: \(0.9165 * 0.5 ≈ 0.45825\).

Adding them together: \(0.3464 + 0.45825 ≈ 0.80465\). So approximately 0.8047.

Now, moving on to the second term: \(\sin(\beta - \alpha)\). Using the sine subtraction formula:

\[\sin(\beta - \alpha) = \sin \beta \cos \alpha - \cos \beta \sin \alpha.\]

Again, substituting the known values:

\[\sin(\beta - \alpha) = (0.5)(0.9165) - (0.8660)(0.4).\]

Calculating each term:

First term: \(0.5 * 0.9165 ≈ 0.45825\),
Second term: \(0.8660 * 0.4 ≈ 0.3464\).

Subtracting them: \(0.45825 - 0.3464 ≈ 0.11185\). So approximately 0.1119.

Now, multiplying the two results together:

\(0.8047 * 0.11185 ≈ 0.0906\).

Wait, but this seems a bit low. Let me check my calculations again to ensure I didn't make a mistake.

First, let's verify the values of \(\cos \alpha\) and \(\cos \beta\):

\(\cos \alpha = \sqrt{1 - 0.4^2} = \sqrt{0.84} ≈ 0.9165\). Correct.

\(\cos \beta = \sqrt{1 - 0.5^2} = \sqrt{0.75} ≈ 0.8660\). Correct.

Then:

\(\sin(\alpha + \beta)\):

\(0.4 * 0.8660 = 0.3464\),
\(0.9165 * 0.5 = 0.45825\),
Sum: 0.3464 + 0.45825 = 0.80465. That's correct.

\(\sin(\beta - \alpha)\):

\(0.5 * 0.9165 = 0.45825\),
\(0.8660 * 0.4 = 0.3464\),
Difference: 0.45825 - 0.3464 = 0.11185. Correct.

Multiplying 0.80465 * 0.11185:

Let me compute this more accurately:

0.8 * 0.1 = 0.08,
0.8 * 0.01185 ≈ 0.00948,
0.00465 * 0.1 ≈ 0.000465,
0.00465 * 0.01185 ≈ 0.000055.

Adding these up:

0.08 + 0.00948 = 0.08948,

0.000465 + 0.000055 ≈ 0.00052.

Total ≈ 0.08948 + 0.00052 = 0.09.

So approximately 0.09. Hmm, so the product is approximately 0.09. But the question is asking for an exact value, I suppose, or perhaps a simplified expression. Let me think again.

Wait, maybe there's a trigonometric identity that can be applied here to simplify the product before calculating. Let me recall that:

\[\sin(A + B)\sin(A - B) = \sin^2 A - \sin^2 B.\]

Wait, is that correct? Let me verify.

Using the identity:

\[\sin(A + B)\sin(A - B) = (\sin A \cos B + \cos A \sin B)(\sin A \cos B - \cos A \sin B) = (\sin A \cos B)^2 - (\cos A \sin B)^2 = \sin^2 A \cos^2 B - \cos^2 A \sin^2 B.\]

Hmm, that's not the same as \(\sin^2 A - \sin^2 B\). Let me see:

Alternatively, perhaps using another identity. Let me recall that:

\[\sin(A + B)\sin(A - B) = \sin^2 A - \sin^2 B.\]

Wait, let me check with specific angles. Let’s take A = 45°, B = 30°. Then:

\(\sin(75°)\sin(15°) = \sin(75°)\sin(15°)\). Let's compute:

\(\sin 75° ≈ 0.9659\), \(\sin 15° ≈ 0.2588\). Product ≈ 0.9659 * 0.2588 ≈ 0.25.

Meanwhile, \(\sin^2 45° - \sin^2 30° = (0.5)^2 - (0.5)^2 = 0.25 - 0.25 = 0\). Wait, that's not matching. So the identity I thought of is incorrect.

Wait, maybe another identity. Let me recall:

\[\sin(A + B)\sin(A - B) = \sin^2 A - \sin^2 B.\]

Wait, with A = 45°, B = 30°, \(\sin^2 45° - \sin^2 30° = 0.25 - 0.25 = 0\), but actual product is 0.25. So that identity is not correct. So perhaps my initial thought was wrong.

Let me try expanding the product:

\[\sin(A + B)\sin(A - B) = (\sin A \cos B + \cos A \sin B)(\sin A \cos B - \cos A \sin B) = (\sin A \cos B)^2 - (\cos A \sin B)^2.\]

Which is:

\[\sin^2 A \cos^2 B - \cos^2 A \sin^2 B.\]

Alternatively, this can be written as:

\[\sin^2 A (1 - \sin^2 B) - (1 - \sin^2 A)\sin^2 B = \sin^2 A - \sin^4 B - \sin^2 B + \sin^4 A.\]

Wait, that seems complicated. Maybe there's another way to express this.

Alternatively, perhaps using the product-to-sum identities:

\[\sin \alpha \sin \beta = \frac{1}{2} [\cos(\alpha - \beta) - \cos(\alpha + \beta)].\]

But here, we have \(\sin(A + B)\sin(A - B)\), which is similar. Let me apply the product-to-sum identity:

Let’s set \(C = A + B\) and \(D = A - B\), then:

\[\sin C \sin D = \frac{1}{2} [\cos(C - D) - \cos(C + D)].\]

But \(C - D = (A + B) - (A - B) = 2B\), and \(C + D = (A + B) + (A - B) = 2A\). Therefore:

\[\sin(A + B)\sin(A - B) = \frac{1}{2} [\cos(2B) - \cos(2A)].\]

So, substituting back into our original expression, where \(A = \alpha\) and \(B = \beta\), we get:

\[\sin(\alpha + \beta)\sin(\beta - \alpha) = \frac{1}{2} [\cos(2\beta) - \cos(2\alpha)].\]

Therefore, instead of calculating each sine term separately, we can compute this expression directly. Let's compute \(\cos(2\beta)\) and \(\cos(2\alpha)\).

First, we need to find \(\cos(2\alpha)\) and \(\cos(2\beta)\). Using the double-angle formula:

\[\cos(2\theta) = 1 - 2\sin^2 \theta.\]

So,

\[\cos(2\alpha) = 1 - 2(0.4)^2 = 1 - 2(0.16) = 1 - 0.32 = 0.68,\]
\[\cos(2\beta) = 1 - 2(0.5)^2 = 1 - 2(0.25) = 1 - 0.5 = 0.5.\]

Therefore,

\[\sin(\alpha + \beta)\sin(\beta - \alpha) = \frac{1}{2} [0.5 - 0.68] = \frac{1}{2} (-0.18) = -0.09.\]

Wait, but earlier when I calculated using the product of the two approximate sine terms, I got approximately 0.09, which is positive. Now, using the identity, I get -0.09. There's a discrepancy here. That means I must have made a mistake somewhere.

Wait a minute, let's check the identity again. The product-to-sum identity says:

\[\sin C \sin D = \frac{1}{2} [\cos(C - D) - \cos(C + D)].\]

But in our case, \(C = \alpha + \beta\) and \(D = \beta - \alpha\). Therefore:

\[\sin(\alpha + \beta)\sin(\beta - \alpha) = \frac{1}{2} [\cos((\alpha + \beta) - (\beta - \alpha)) - \cos((\alpha + \beta) + (\beta - \alpha))].\]

Simplifying the arguments:

First term inside cos:

\((\alpha + \beta) - (\beta - \alpha) = \alpha + \beta - \beta + \alpha = 2\alpha\).

Second term:

\((\alpha + \beta) + (\beta - \alpha) = \alpha + \beta + \beta - \alpha = 2\beta\).

Therefore,

\[\sin(\alpha + \beta)\sin(\beta - \alpha) = \frac{1}{2} [\cos(2\alpha) - \cos(2\beta)].\]

Wait, so actually it's \(\frac{1}{2} [\cos(2\alpha) - \cos(2\beta)]\), not the other way around. So in my previous calculation, I had the sign wrong. Let me correct that.

Therefore,

\[\frac{1}{2} [\cos(2\alpha) - \cos(2\beta)] = \frac{1}{2} [0.68 - 0.5] = \frac{1}{2} (0.18) = 0.09.\]

Ah, that matches the approximate value I got earlier. So the correct value is 0.09. Therefore, the exact value is 0.09, which is 9/100, or 0.09. So, the answer is 0.09.

But let me verify once more to be certain. Let's compute \(\cos(2\alpha)\) and \(\cos(2\beta)\) again:

\(\cos(2\alpha) = 1 - 2*(0.4)^2 = 1 - 0.32 = 0.68\),
\(\cos(2\beta) = 1 - 2*(0.5)^2 = 1 - 0.5 = 0.5\).

So, \(\cos(2\alpha) - \cos(2\beta) = 0.68 - 0.5 = 0.18\).

Multiply by 1/2: 0.18 * 0.5 = 0.09. Correct.

Alternatively, if I compute \(\sin(\alpha + \beta)\) and \(\sin(\beta - \alpha)\) exactly using the original formula, perhaps we can get an exact expression.

Wait, but the problem is given with 0.4 and 0.5, which are decimal numbers. However, it's possible that the exact value is 0.09, which is 9/100. Let me check if there's an exact expression.

Alternatively, maybe there's a different approach. Let me think.

Another way: Let's compute the product \(\sin(A + B)\sin(B - A)\). Using the identity we derived, it's \(\frac{1}{2} [\cos(2A) - \cos(2B)]\). Which gives us 0.09. But perhaps we can compute it using exact expressions.

Wait, but 0.4 and 0.5 are rational numbers, so maybe there's a way to express this without decimal approximations.

Let me try to compute \(\cos(2\alpha)\) and \(\cos(2\beta)\) exactly.

Given \(\alpha = \arcsin(0.4)\), so \(\sin \alpha = 0.4 = 2/5\). Therefore, \(\cos \alpha = \sqrt{1 - (2/5)^2} = \sqrt{1 - 4/25} = \sqrt{21/25} = \sqrt{21}/5\).

Similarly, \(\beta = \arcsin(0.5)\), so \(\sin \beta = 1/2\), and \(\cos \beta = \sqrt{1 - (1/2)^2} = \sqrt{3}/2\).

Therefore,

\(\cos(2\alpha) = 1 - 2\sin^2 \alpha = 1 - 2*(4/25) = 1 - 8/25 = 17/25 = 0.68\),

\(\cos(2\beta) = 1 - 2\sin^2 \beta = 1 - 2*(1/4) = 1 - 1/2 = 1/2 = 0.5\).

Thus,

\(\frac{1}{2} [\cos(2\alpha) - \cos(2\beta)] = \frac{1}{2} [17/25 - 1/2] = \frac{1}{2} [34/50 - 25/50] = \frac{1}{2} [9/50] = 9/100 = 0.09\).

So, exactly 9/100 or 0.09. Therefore, the answer is 0.09.

But to make sure, let's compute the original expression without using the identity, but directly using the sine addition and subtraction formulas.

Compute \(\sin(\alpha + \beta)\):

\(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta = (2/5)(\sqrt{3}/2) + (\sqrt{21}/5)(1/2)\).

Simplify:

First term: (2/5)(√3/2) = (√3)/5,

Second term: (√21)/5 * 1/2 = √21/10.

So,

\(\sin(\alpha + \beta) = √3/5 + √21/10 = (2√3 + √21)/10\).

Similarly, compute \(\sin(\beta - \alpha)\):

\(\sin(\beta - \alpha) = \sin \beta \cos \alpha - \cos \beta \sin \alpha = (1/2)(√21/5) - (√3/2)(2/5)\).

Simplify:

First term: (√21)/10,

Second term: (√3)/5.

So,

\(\sin(\beta - \alpha) = √21/10 - √3/5 = √21/10 - 2√3/10 = (√21 - 2√3)/10\).

Now, multiply the two results:

[(2√3 + √21)/10] * [(√21 - 2√3)/10].

Let me compute this product:

Multiply numerator: (2√3 + √21)(√21 - 2√3).

Using the distributive property:

First term: 2√3 * √21 = 2√(3*21) = 2√63 = 2*3√7 = 6√7,

Second term: 2√3 * (-2√3) = -4*(√3)^2 = -4*3 = -12,

Third term: √21 * √21 = (√21)^2 = 21,

Fourth term: √21 * (-2√3) = -2√(21*3) = -2√63 = -6√7.

Adding all terms together:

6√7 -12 +21 -6√7 = (6√7 -6√7) + (-12 +21) = 0 +9 =9.

Therefore, the numerator is 9, and the denominator is 10*10=100. Thus, the product is 9/100 = 0.09. Exactly the same result.

Therefore, the exact value is 9/100, which is 0.09. So, the final answer is 0.09.

But just to make sure, let's verify once more using decimal approximations:

Compute \(\sin(\alpha + \beta)\):

First, \(\alpha = \arcsin(0.4)\). Let's compute this angle in radians:

\(\arcsin(0.4) ≈ 0.4115\) radians,

\(\beta = \arcsin(0.5) = \pi/6 ≈ 0.5236\) radians.

Thus, \(\alpha + \beta ≈ 0.4115 + 0.5236 ≈ 0.9351\) radians,

\(\sin(0.9351) ≈ 0.8040\).

\(\beta - \alpha ≈ 0.5236 - 0.4115 ≈ 0.1121\) radians,

\(\sin(0.1121) ≈ 0.1118\).

Multiplying them: 0.8040 * 0.1118 ≈ 0.0906. Which is approximately 0.0906, close to 0.09. The slight discrepancy is due to rounding errors in the approximate angle measures.

But given that when we compute it exactly, we get 0.09, which is 9/100, so the exact answer is 0.09.

Therefore, the final answer is \boxed{0.09}.

**Final Answer**
\boxed{0.09}
Therefore, the final answer is $\boxed{0.09}$.
</think>