Why didn't Hans Gruber know what Mr. Takagi looked like?

Though the movie does not provide clear evidence about whether Hans had recognized Takagi on sight, I have always assumed that Hans had a good idea which man was Takagi, but had to confirm, hence the dialogue.

It seems likely that Hans would have looked at a picture (or pictures) of Takagi in advance -- but looking at a picture doesn't guarantee he would have easily recognized Takagi when meeting him for the first time, in a group.

In any case, the dialogue also happens to do a great job developing Hans's character: it reveals he is (1) European, (2) an intelligent speaker, (3) thorough in his preparation, and (4) confidently in charge of the current situation.

So from Hans's perspective, the dialogue is indeed a chilling and bad-ass way to introduce himself to everyone at the party he just crashed. And similarly, from the screenwriters' perspective, the dialogue is an ideal introduction of Hans to the audience.

This was clearly a scare tactic and a show of force and knowledge. If you notice in that scene, he walks next to every Asian man in the group except Takagi, which allows him to complete his biography before identifying his target.

Considering the precision with which the preceding events were carried out, indicating a vast knowledge of the target (including building schematics), it's unfathomable that they wouldn't have known what Takagi looked like.

One brief point in addition to the answers above: Die Hard takes place in 1987 or 1988, before social media, the web, 24 hour cable news, or even the desktop publishing revolution. For someone raised in the modern era it can be hard to imagine how scarce rich media information for a non-celebrity was in the late '80s. It's not only plausible but quite reasonable that Gruber could have gained the information he quotes (which reads to me like a bio from a stolen investor prospectus) without coming across a photo of Takagi.

There would have been three likely sources for such a picture: A magazine, a newspaper, or a tv broadcast. Each of these forms of exposure would have been strenuously avoided by a Japanese executive in that era. Even for an executive of a U.S. company--especially in the finance industry--publicity seeking was still frowned upon. As a young executive at Bain Capital, Mitt Romney was nearly fired for allowing a photo of himself to appear in a magazine.

Your conjecture is correct, it was a scare tactic. By getting a chance to show his encyclopedic knowledge of Takagi, Gruber is showing that his team are both there for a specific reason, and are well organised. They are not just some random thugs on an opportunistic crime.

Given what Hans Gruber he wanted from Takagi (the password), this was a good tactic for establishing that he was in charge and in the know. It was a chilling way of telling, "Think twice about trying to bluff me." It is the same reason he mentions Takagi's suit. He also uses it to cow everyone else in the room.

But it was also a great way of sizing up Takagi. How far through his speech would Takagi let him get? How rattled would Takagi by the time he was done? Would Takagi identify himself at all?

Of course, Takagi was a badass. He was trying to size up Hans as well. When he comes forward, he does it completely unfazed. All the more reason to be upset when Hans shoots him.

But did Takagi actually know the password?

I'm in agreement with the others. Hans Gruber's monologue was simply a movie plot device to intimidate, assert control and scare the living daylights out of everyone in the room. Lest some half-drunken party reveler get the not-too-smart idea to try to take on a bunch of armed hostage takers. It shows who is really in charge, and it is not Mr Takagi anymore, and to show how well-prepared he was for any eventuality.

Today, people put pictures of CEOs up on company websites or social media. Prior to the age of the internet, pictures where a rare and private thing. Having your picture be easy to find basically meant you where a celebrity.

All of the information he displayed could have been gathered without personally stalking Takagi.

At that date, it is perfectly reasonable that the only way to get a picture of Takagi would be to hire someone to stalk him; hire a PI. Such a PI could end up "showing their hand" in that they would have to ask about Takagi to other people. It is a risk.

Alternatively, they could create a fake business for which Takagi would meet them. But here again this risks exposure.

Learning about Takagi from public information is risk free. Get newspaper archives from the other side of the planet and read them. Never tell the librarian you are looking for information on Takagi in particular, and don't use logged newpaper rooms, and there is no paper trail connecting you to Takagi or Nakatomi.

Prior to the internet, pictures where distributed by hand or by newspapers and magazines. Getting a picture of something took a lot of effort.

The research that could have tipped off their target would be the blueprints for the building. That probably involved some risk; but the CEO of a large corporation might actually have private security looking for stalkers, while the architecture firm that has the blueprints, the municiple office of records, or the like is far less likely to have.

A core part of the plan was utter and complete surprise on the part of the target. Nothing that could tip off the target that wasn't key to the mission was permitted. Finding out who Takagi was was a solved problem; threaten to kill people who could not be him until he gives himself up. Actually stalking him to get a picture would be a risk.