This is a general essay plan that you could use to answer the question (which was in fact a previous past exam question). I used the mark scheme, which included events which will be likely to be mentioned, as a spine of my essay. Hope it helps!
Point: It could be argued that the continual push from both superpowers to develop their nuclear arsenal's during the period 1949-63 inevitably shows how the arms race did little to restrain their actions.
Evidence: 1949- Soviets develop Nuclear bomb. 1952: following the Mike Tests in the US develop the H-Bomb. 1953- Russia develop their own H-Bomb following the Joe 4 testing. ICBM's developed by the late 1950s. US had 18,000 warheads by 1960.
Argue: both sides were desperate to be in the lead of the arms race. Showed how the competitive nature of the Cold War and the arms race in particular did little to restrain the actions of the US.
Counter Argue: It could be argued that the fact because there was nuclear parity in the early stages of the arms race, it actually deterred conflict and therefore restrained their actions. This can be referred to as the notion of mutually assured destruction (MAD)
Evaluate: The counter argument holds great strength as the US refused to use a nuclear bomb in the Korean War and also refused to aid the Hungarian rebels, despite their pleas in 1956 as the fear of nuclear war, which may have been induced by the US entering a Soviet sphere of influence, restrained their actions. Therefore providing evidence to dispute the claim made in the title.
Point: The cost of nuclear war did restrain the actions of the superpowers. The Russian economy was faultering and therefore the USSR sought a better relationship with the US so they could reduce…