Final answer:
The red stripe on the dress blue trousers of officers and NCO's signifies the valor at the Battle of Chapultepec in the Mexican-American War, commemorating the Marine Corps' sacrifices.
Step-by-step explanation:
The red stripe on the dress blue trousers of officers and NCO's in the U.S. military represents the Battle of Chapultepec during the Mexican-American War in 1847. This battle was notable for its high degree of valor and heroism shown by U.S. forces, particularly by the Marine Corps, which sustained significant casualties. The red stripe, or 'blood stripe', is a distinct feature of the Marine Corps dress uniform and commemorates their sacrifice during that campaign. It is a symbol of honor and respect, paying tribute to those who have boldly led and fought in past conflicts.