while solving questions where we have to prove that LHS=RHS, can we operate on both the sides simultaneously or is it like we can operate on only 1 side?

While solving questions where we have to prove LHS = RHS, we cannot operate both the sides simulataneously, since that is what we have to prove.

 

Example: To prove (a - b)² =a² - 2ab + b²

We first have to either take LHS and prove it equal to the RHS or take the RHS and prove it equal to the LHS.

You can also take the LHS, and arrive at some expression, and then consider the RHS, and arrive at the same expression to prove the required.

But both, LHS and RHS should be operated separately.