I think I've seen a similar occurance earlier, but it's literally years ago. Here's a couple of things you could try / verify:
1. Check that you have all the files needed in "C:\Program Files (x86)\GogTasks" .List of files here:
link2. Check that the version of GogTasks2010.dll is 6.3.7 på right clicking on it ,selecting "properties" then Details
3. Open REGEDIT.EXE, navigate to "HKEY_CURRENT_USER\Software\Microsoft\Office\Outlook\Addins\GogTasks". Check that the values are as below.
(Standard)="3"
"Manifest"="C:\Program Files (x86)\GogTasks\GogTasks2010.vsto|vstolocal"
"FriendlyName"="GogTasks"
"Description"="Google Tasks synchronization service"
"LoadBehavior"= 0x00000003
Finally you're saying "Outlook doesn't show any addin" -does that mean
any addin? What does
this screen show, if anything? Any active addins? Any inactive?