It's an emphasis of questioning. You can absolutely delete “究竟” and the sentence still holds correctly. Another similar word with "究竟" is "到底". Hope it helps.

究竟 的相关例句 1. It is difficult to convey the sheer complexity of the situation. 很难说清楚形势究竟有多复杂. 2. Few people knew exactly what the rebels planned to do. 没有人知道叛军究竟打算做什么. 3. Where under the sun could I have put the purse? 我究竟把钱包放在哪儿了? 4. You'd better decide for yourself whether to go or not. 究竟去不去,你自己拿主意吧. 5. I'm not sure just what day he will leave. 我不敢说他究竟哪一天去. 6. How, and with what result , you shall presently see. 究竟如何, 下回分解. 7. " And what does HE want?'she growled. “ 他究竟要 什么 ? ” 她大声吼道. 8. Whoever could have done such a dreadful thing? 究竟是谁能干出这种可怕的事 呢 ? 9. Whatever planted that idea in your head? 究竟是什么使你产生了这种念头? 10. The vastness of space defied comprehension. 外层空间的茫茫无垠让人无法弄清其究竟. 来自《简明英汉词典》

是反问的一种吧。意思可能是，谁教得好比较难说。

jiū jìng 究竟 （1）用在句尾，表示：结果（outcome）; 原因 例如：如果要罗比做事,他总爱问个究竟。（为什么要做这件事Why to do this thing） Robbie always asks the whys and wherefores when he is told to do something. （2）到底exactly; after all 例如：你究竟（到底）在干什么？What ever are you doing? 它究竟（到底）在哪儿?Where the dickens is it?