This answer depends on if this is a free response question or if it is just an A or B multiple choice. Assuming it is free response.
Technically the space race and nuclear programs during the cold war were the same thing called an "Arms Race" both the US and the USSR were both innovating to make their ICBM's (Intercontinental Ballistic Missile) better and faster than the other. To flex their military muscles they often used similar missiles to launch things into space to prove who had the strongest and more capable scientists, engineers, etc.. So The Space race and Nuclear programs were essentially the same thing.
But, if you have to pick, Nuclear Programs would be the answer. Both nations wanted to be more powerful than the other, both fearing each other's ideals and governments, so as long as the US thought the USSR had spies and plans to attack the US, they were going to increase military spending on special weapons projects, to create more powerful nuclear weapons. Lots of other things contributed to the end of the cold war, but a small reas9n is that fact for Mutual Destruction, which basically means if they nuke us, we'll nuke them.
