1. The problem is to verify the trigonometric identity involving the sine of sums and differences. 2. We start with the expression $$\sin\frac{A+B}{2}$$ and substitute it as $$\sin\frac{\pi - C}{2}$$. 3. Using the identity $$\sin\left(\frac{\pi}{2} - x\right) = \cos x$$, we rewrite: $$\sin\left(\frac{\pi}{2} - \frac{C}{2}\right) = \cos\frac{C}{2}$$. 4. Therefore, the substitution shows that: $$\sin\frac{A+B}{2} = \cos\frac{C}{2}$$. 5. This is a standard trigonometric identity that relates sine and cosine through complementary angles. 6. The key rule used is the complementary angle identity for sine and cosine. 7. This substitution is useful in simplifying expressions involving sums of angles. Final answer: $$\sin\frac{A+B}{2} = \cos\frac{C}{2}$$