We can simplify the expression \(\sin\left(\frac{5\pi}{12}\right)\cos\left(\frac{\pi}{12}\right) + \cos\left(\frac{5\pi}{12}\right)\sin\left(\frac{\pi}{12}\right)\) by recognizing that it resembles the sine angle addition formula:
\[ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \]
In this case, we can let:
- \(a = \frac{5\pi}{12}\)
- \(b = \frac{\pi}{12}\)
Then the expression can be written as:
\[ \sin\left(\frac{5\pi}{12} + \frac{\pi}{12}\right) \]
Now, we need to compute \(\frac{5\pi}{12} + \frac{\pi}{12}\):
\[ \frac{5\pi}{12} + \frac{\pi}{12} = \frac{5\pi + \pi}{12} = \frac{6\pi}{12} = \frac{\pi}{2} \]
Thus, the expression simplifies to:
\[ \sin\left(\frac{\pi}{2}\right) \]
Knowing that \(\sin\left(\frac{\pi}{2}\right) = 1\), we conclude:
\[ \sin\left(\frac{5\pi}{12}\right)\cos\left(\frac{\pi}{12}\right) + \cos\left(\frac{5\pi}{12}\right)\sin\left(\frac{\pi}{12}\right) = 1 \]
Thus, the simplified expression is:
\[ \boxed{1} \]