You have three likely issues, two are certain and I will start with those
1) You have a new tank that is not cycled enough for any fish, let alone sensitive fish like tangs. Any ammonia at all can and does kill, nitrite is less toxic but still indicative of an immature tank. The way that your tank looks is also an indicator that you're still cycling. It's likely to get worse before it gets better, aesthetically.
2) Your tank is WAY too small for even a single small tang, let alone two decent-sized tangs. They swim vast distances in the wild, as mentioned. A stressed fish can succumb to nearly anything, and these conditions would without a doubt be immensely stressful with the cramped space and uncycled water
3) It's possible that you are also dealing with parasites, perhaps even likely. I assume you've not quarantined your fish, and with these other two immense stress factors nearly any parasite infestation could very easily get the upper hand. Flukes, ich, velvet, uronema, and infections are common. I don't see any obvious symptoms of any of these in the photos, however with the immense levels of stress present it's not likely that the parasite would have to advance past the gill infestation stage before the fish perishes. In other words, in a remotely healthy or less stressed fish, it would eventually get spots if it had ich or velvet on the exterior of the fish, typically. In these conditions it wouldn't need to advance to that stage to be fatal, most likely.
I'd rehome the fish and avoid angelfish/tangs or any kind for a tank of that size. Slow down and do your research, don't rely on any LFS (particularly the one that sold you thsoe fish) for information. Some good LFS exist but many have employees that haven't a clue, don't care, or simply just want to make the sale and aren't worried about the aftermath.
Suitable tankmates would include gobies and the like.