It is possible to download TaiG 2.3.1 jailbreak for iOS 8.4 via direct link provided in this post below on your device.
As an interesting move has happened recently. The Chinese team TaiG have launched a new version of their iOS 8.4 jailbreak tool for Windows, but have labeled it as a beta. Version 2.3.1 beta gains the latest version of Cydia, version 1.1.20, and that’s about all, nothing new at all.
According to the change log, no other changes were brought to the TaiG tool outside of the Cydia update. For such a small change, it’s weird that they’ve labeled it as being a beta, but maybe this will become a new trend going forward until they have enough successful testers to deem it a stable release in future.
That's why TaiG team commented that it is not necessary to use the new tool, since it’s in beta. So TaiG isn’t recommending that users shouldn't use this beta version and use the previous public version of the jailbreak tool - version 2.3.0 to jailbreak their devices.
Indeed, it looks as if they’re soliciting feedback from those who use the tool. In the change log for 2.3.1 beta TaiG notes that the latest update integrates the Cydia 1.1.20.
That's way if you plan to jailbreak your device running iOS 8.4 the previous version is what you should use, not the recent update.
Once jailbroken, you can install the latest version of Cydia Installer by simply launching Cydia and hitting the refresh button.
But if you want to use the latest update then via this direct link you can download TaiG 2.3.1 jailbreak for iOS 8.4.
After this TaiG beta launch we can say that we all should expect future TaiG release, even major one for new firmware builds like iOS 9. It sounds really great, isn't it?