The soldier gave his life ________ the student’s life in the

选择 A.to save
解析：
The soldier gave his life to save the student's life 中,to save the student's life 是 The soldier gave his life 的目的状语,即“为了救那个学生,这个战士献出了自己的生命”.
The soldier gave his life to saving the student's life 中,to 是介词,后面的 saving the student's life 是动名词短语,套用的是 give one's life to sth.（把生命献给某事/为某事捐躯）句型,即“这个战士救那个学生而捐躯”.
从语法上讲,第一种说法没有问题,第二种似乎也成立,但是从情理上讲,后一个句子属于逻辑悖论.可以把生命献给比生命还要重要的事业,比如说,give one's life to the modernization construction（把生命献给现代化建设）,give one's life to one's motherland（把生命献给祖国）,但是不能献给某个具体行为.因此,只能说 give one's life to do sth.或者 give one's life to sth.而不能说 give one's life to doing sth.

A吧
He would give his life to keep the bridge safe.
就是牺牲生命，他也要保住大桥。
Do not fear… He would give his life to save our world.
不用害怕…为了拯救我们的世界，他愿意付出自己的生命。

a A