$\frac{2^{2003} \cdot 9^{1001}}{4^{1001} \cdot 3^{2003}}+\frac{2^{2002} \cdot 9^{1001}}{4^{1001} \cdot 3^{2003}}$

Step 1 ：Rewrite the given expression as: \(\frac{2^{2003} \cdot 3^{2002}}{2^{2002} \cdot 3^{2003}}+\frac{2^{2002} \cdot 3^{2002}}{2^{2002} \cdot 3^{2003}}\)

Step 2 ：Factor out the common terms in the numerator and denominator: \(\frac{2^{2002} \cdot 3^{2002} (2)}{2^{2002} \cdot 3^{2003}}+\frac{2^{2002} \cdot 3^{2002}}{2^{2002} \cdot 3^{2003}}\)

Step 3 ：Cancel out the common terms: \(\frac{2}{3}+\frac{1}{3}\)

Step 4 ：Add the fractions: \(\frac{2}{3}+\frac{1}{3} = \frac{3}{3}\)

Step 5 ：\(\boxed{1}\)