>There are lots of ways of
>solving the same problems.

>Then
>why do Mathematicians still try
>to figure out some other
>ways?

If they did not, there would not be lots of ways of solving the same problems in the first place, right?

>Why do some people spend most
>of their time to do
>something that's already done?

I do not know about its being most of their time, but certainly many people spend a lot of time proving already established theorems.

Why? Because people seek enlightenment, not just truth. Often the first proof is too obscurred to shed light as to why something is true.

Some proofs are better than others - simpler, clearer, more transparent, more general, etc. Mathematicians seldom prove something just in order to prove, exercise their brain or build a reputation. What they prove is often a part of a more general scheme. Multiple proofs help them see a bigger picture.