I have heard that when White makes defensive moves, Black should take the center and be aggressive.
In such a case, is there a problem with Black making defensive moves too? Is that a bad idea?
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Being aggressive and taking the center are two different things.
Regarding being aggressive, if White is making defensive moves, it's OK for Black to do the same as long as each move has a strong purpose. For example, Black might fix flaws in his position by repositioning poorly placed pieces or securing his King. If he instead chooses to attack and there are exploitable flaws, his attack can he blunted.
For example, if Black's King is exposed yet he attacked, White would have the opportunity to gain a tempo by moving an endangered piece and checking the King. This is like giving White two moves in a row on the area of the board where the attack is happening. Or, worse, White could play a Zwischenzug. Both these things things make calculation difficult. Better to first move the King to a safer square.
Taking the center, which is less aggressive, is rarely a bad idea, as long as one can hold onto the real estate. Taking the center is good preparation for attack and gives one the flexibility in choosing how the attack should be conducted.
Yes, when your opponent makes defensive and passive moves, you should try to take initiative and attack, no matter what color you are playing.
Because White make move first, in very first stage of game they have initative, and Black must counteract against it, so they technically defense, but after opening this advantage becomes a little significant.
There is only one difference between black and white: The first move(no, the color does not count :p).
This means that white starts with the initiative. They can make threats first and black should respond. At the moment white lets the initiative go(that is, allows black to make threats to which they must respond) the colors have "switched"(no, this does not mean the pieces actually change color. It means their state have changed).
The initiative in chess is very important. You feel it everytime you wish that "if only it was my turn to move" :).