It’s frustrating when this happens. I’ve occasionally experienced it as well.
The issue here is that it’s typically not an issue that’s in our control. It’s usually the website’s fault.
So, what to do? I only have a few suggestions here.
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It’s the website’s fault
First, if the website that you’re on is telling you that you need IE8 when you’re already running IE8 and you’re not playing any games trying to trick the website somehow (which I imagine is unlikely in a situation like this), then… it’s not your fault. It’s the website’s fault. They’re misidentifying the browser that you are using.
Now, when that happens, one thing that I would have you do first is hit Refresh. In other words, hit the F5 key on your keyboard to reload the page immediately. Internet Explorer will sometimes then switch into what it calls Compatibility Mode and that may allow the page to be fetched differently (I’m honestly not sure).
Try a different browser
It’s also possible that the website is detecting the version of Internet Explorer incorrectly. In this case, it’s not your issue or Internet Explorer’s issue; it’s simply the website that you’re visiting that’s misidentifying the browser.
In this case, the best approach may very well be to just forget about Internet Explorer completely. Go download a copy of Chrome or Firefox; install one of those (either one will do) and use that as your browser. That completely bypasses this issue because most of the sites will then bypass all of the Internet Explorer version checking shenanigans and work just fine.
My bottom line recommendation: switch browsers. The problem will just go away. I run Chrome for most things, and I’ve not seen this problem since I switched.