In this problem, we are given:

1. The equation $A + B + C = \frac{\pi}{2}$.
2. The expression to evaluate or simplify: $\sin^2 A + \sin^2 B + \sin^2 C$.

Let's proceed with solving this.

Since $A + B + C = \frac{\pi}{2}$, we can consider a trigonometric identity involving complementary angles. We know:

1. If $C = \frac{\pi}{2} - (A + B)$, then $\sin C = \cos(A + B)$.
2. We could use trigonometric identities to express $\sin^2 A + \sin^2 B + \sin^2 C$ in terms of $A$ and $B$.

Expanding $\sin^2 A + \sin^2 B + \cos^2 (A + B)$ (as $\sin C = \cos(A + B)$) will likely simplify based on trigonometric identities. This involves a known identity:

$\sin^2 A + \sin^2 B + \cos^2 (A + B) = 1$

Thus, for $A + B + C = \frac{\pi}{2}$, the value of $\sin^2 A + \sin^2 B + \sin^2 C$ simplifies to 1.

Would you like further details on trigonometric identities, or do you have any specific questions about this solution?

Here are five related questions to expand on this concept:

1. How does the Pythagorean identity apply to complementary angles?
2. Can this problem be solved using vector representation in trigonometry?
3. What are other identities involving $\sin^2$ and $\cos^2$ terms?
4. How does this identity change if the sum is different from $\frac{\pi}{2}$?
5. What would be the value if $A + B + C = \pi$ instead?

Tip: Remember that complementary angle identities are especially useful in simplifying expressions where the angles sum to $\frac{\pi}{2}$ or $\pi$.