1. 让步状语从句，可以经常省略系动词，连同情态动词一起省略(may be)：

whatever the problem being tackled = whatever the problem being tackled may be 

2. being tackled，是现在分词作定语，修饰the problem，意为“正在被解决的”。

3. 让步从句把主句中的谓语和宾语分隔，现在还原回去：

A successful scientist needs to deeply believe, whatever the problem being tackled, that there is always a better way to approach that problem than the path currrently being taken. 等于：

Whatever the problem being tackled may be, a successful scientist needs to deeply believe that there is always a better way to approach that problem than the path currrently being taken.

【翻译】无论正在解决的问题是什么，一位成功的科学家都需要坚信（深信），总有一个比目前采用的途径更好的方法，来处理那个问题。