A. Why I believe Cat knows that Kara is Supergirl
1. Alex and Eliza
When Cat figured out that Kara was Supergirl, Cat investigated Kara's life very deeply, even going so far as to get her report cards from grade school. There is no way Cat could have investigated Kara's life so thoroughly without seeing pictures of Alex and Eliza.
In the final episode of the first season, Cat ended up in an old television station with Max, J'onn, Supergirl, Alex, and Eliza. The notion that Cat wouldn't have recognized them is simply absurd. I consider this to be absolute proof that Cat knows Kara is Supergirl.
2. James, Winn, and Barry
Cat knows that Kara, James, and Winn are close friends. In the episode "Falling," it is clear that Cat also knows that James and Winn are close friends with Supergirl. In the episode "World's Finest," Cat reveals that she knows Barry is the Flash, and she must realize that the Flash's association with Kara, James, and Winn while simultaneously working with Supergirl could not have been a coincidence.
This is not definitive, but it seems absurd that Cat would fail to realize that Kara is connected to the same people Supergirl is connected to.
3. Bizarro
During Kara's date with Adam, Bizarro kidnapped her. Cat never asked why Bizarro would target Kara (which, since Cat is a journalist, is simply stunning in itself), nor did Cat seem surprised at Kara's return. This is the opposite of the way a person would normally react upon hearing about a friend being abducted by a supervillain.
Again, this is not definitive, but it is extremely telling.
B. Cat's behavior
This means Cat is pretending not to know Kara's secret, which I think is perfectly within character. But this raises two questions:
1. How does Cat reconcile this knowledge with the fact that she saw Kara and Supergirl standing next to each other?
2. If Cat knows Kara is Supergirl, why didn't Cat continue to threaten to fire Kara like she did immediately after originally figuring that out?
The first question is easy to answer. Cat knows perfectly well that aliens with many strange powers exist, and that Supergirl works with a government agency which uses highly advanced technology. Cat may not know exactly how Kara performed the illusion, but she would have no trouble believing that it was an illusion.
I find the second question much more interesting, and I believe I have an answer for that, as well.
Cat is sometimes inconsistent. In some episodes, she is extremely health-conscious; in others, she's downing boxes of cupcakes and drinking fattening drinks because life is short. After Kara broke up with Adam, Cat insisted that her and Kara's relationship should be strictly professional; several episodes later, Cat freely advised Kara on how to land James as a boyfriend and this after sharply criticizing Kara earlier in the season for throwing herself at James. Therefore, I conclude that Cat often changes her mind without admitting it and without explaining it, and I think she could easily do the same with her attitude towards Kara's secret.
C. My theory
When Cat figured out Kara was Supergirl, she wanted to fire Kara because A) superheroes should not lower themselves to the status of fetching coffee, and B) Supergirl's time could be much better served by helping people every possible moment. When Cat delivered her ultimatum, she assumed Kara would fold and admit that she was Supergirl.
But Cat was wrong. Kara stuck to her guns and gave Cat an impassioned speech, telling Cat how much she loved her job. Cat also stuck to her guns, so Kara stuck to her guns even further and resigned.
In that moment, Cat realized she had gone too far, and that she needed to back off and let Supergirl be Supergirl. But Cat couldn't back down from her ultimatum because she can't afford to lose face. That was why she told Kara, "You could at least finish out the day." Cat sounded snide, but it was obvious that she didn't actually want Kara to leave.
So when Kara performed her illusion, Cat rolled with it. It gave her a way to back down from her ultimatum without losing face. She even said that Kara and Supergirl looked nothing alike, which is simply absurd.
This would also explain why Cat seemed to forgive Kara so easily for breaking up with Adam. She may have realized that Kara pushed Adam away in order to protect him. As Adam's mother, she would have appreciated this very much.
1. Alex and Eliza
When Cat figured out that Kara was Supergirl, Cat investigated Kara's life very deeply, even going so far as to get her report cards from grade school. There is no way Cat could have investigated Kara's life so thoroughly without seeing pictures of Alex and Eliza.
In the final episode of the first season, Cat ended up in an old television station with Max, J'onn, Supergirl, Alex, and Eliza. The notion that Cat wouldn't have recognized them is simply absurd. I consider this to be absolute proof that Cat knows Kara is Supergirl.
2. James, Winn, and Barry
Cat knows that Kara, James, and Winn are close friends. In the episode "Falling," it is clear that Cat also knows that James and Winn are close friends with Supergirl. In the episode "World's Finest," Cat reveals that she knows Barry is the Flash, and she must realize that the Flash's association with Kara, James, and Winn while simultaneously working with Supergirl could not have been a coincidence.
This is not definitive, but it seems absurd that Cat would fail to realize that Kara is connected to the same people Supergirl is connected to.
3. Bizarro
During Kara's date with Adam, Bizarro kidnapped her. Cat never asked why Bizarro would target Kara (which, since Cat is a journalist, is simply stunning in itself), nor did Cat seem surprised at Kara's return. This is the opposite of the way a person would normally react upon hearing about a friend being abducted by a supervillain.
Again, this is not definitive, but it is extremely telling.
B. Cat's behavior
This means Cat is pretending not to know Kara's secret, which I think is perfectly within character. But this raises two questions:
1. How does Cat reconcile this knowledge with the fact that she saw Kara and Supergirl standing next to each other?
2. If Cat knows Kara is Supergirl, why didn't Cat continue to threaten to fire Kara like she did immediately after originally figuring that out?
The first question is easy to answer. Cat knows perfectly well that aliens with many strange powers exist, and that Supergirl works with a government agency which uses highly advanced technology. Cat may not know exactly how Kara performed the illusion, but she would have no trouble believing that it was an illusion.
I find the second question much more interesting, and I believe I have an answer for that, as well.
Cat is sometimes inconsistent. In some episodes, she is extremely health-conscious; in others, she's downing boxes of cupcakes and drinking fattening drinks because life is short. After Kara broke up with Adam, Cat insisted that her and Kara's relationship should be strictly professional; several episodes later, Cat freely advised Kara on how to land James as a boyfriend and this after sharply criticizing Kara earlier in the season for throwing herself at James. Therefore, I conclude that Cat often changes her mind without admitting it and without explaining it, and I think she could easily do the same with her attitude towards Kara's secret.
C. My theory
When Cat figured out Kara was Supergirl, she wanted to fire Kara because A) superheroes should not lower themselves to the status of fetching coffee, and B) Supergirl's time could be much better served by helping people every possible moment. When Cat delivered her ultimatum, she assumed Kara would fold and admit that she was Supergirl.
But Cat was wrong. Kara stuck to her guns and gave Cat an impassioned speech, telling Cat how much she loved her job. Cat also stuck to her guns, so Kara stuck to her guns even further and resigned.
In that moment, Cat realized she had gone too far, and that she needed to back off and let Supergirl be Supergirl. But Cat couldn't back down from her ultimatum because she can't afford to lose face. That was why she told Kara, "You could at least finish out the day." Cat sounded snide, but it was obvious that she didn't actually want Kara to leave.
So when Kara performed her illusion, Cat rolled with it. It gave her a way to back down from her ultimatum without losing face. She even said that Kara and Supergirl looked nothing alike, which is simply absurd.
This would also explain why Cat seemed to forgive Kara so easily for breaking up with Adam. She may have realized that Kara pushed Adam away in order to protect him. As Adam's mother, she would have appreciated this very much.