What are cons of force a web site viewed in IE to compatible mode? Say we force IE9 to IE8 compatiblity mode?
We run legacy web app which is developed since 2000 so it's a mess ball fighting to be compatible with Chrome, Opera, Firefox, IE6/7/8 and now we decide to add IE9 to the list. But with IE9 we run in issues with printing, "Permission deniend" JavaScript errors (probably something about cross-frame JavaScript calls) and next issues - the easy workaround is to force IE9 to behave as a IE8 and then everything works fine. But I am still not sure if it's way to go...
first our app is public site (for our clients)
You have a public website developed in 2000 and it doesn't work on modern browsers? Deprecate it or re-write it.
Don't hack your code to support modern browsers, the website is clearly poorly written and doesn't apply to standards. You can't get away with this.
The only place where you can get away with this level of incompatibility is intranet applications and even then you should simply state "it works on browser X, live with it"
You can't say that to public facing clients. I mean you can try, but have fun losing business to your competitors.
Re-develop your website to match the W3C HTML/CSS standards and the ES5 standards and it will be completely future facing (for some years).
Alas, the way the web works is that anything more then 5 years old is deprecated. So either re-write it every 5 years or get out of the web business.
In terms of actually using compatibility mode, don't. IE6-8 are horrible engines and should be avoided like the plague. If you use them then you can't write future facing standards compliant code.
Your code needs to match the standards and you should fix / shim / patch any browser specific bugs where those browsers don't implement the standards.