If you've trusted your computer but still see the prompt for your Sesame, locate the LastPass data folder on your system, and delete all the files found in that folder.  Then go through the steps to retrust.

Here is a list of data folder locations based on browser and operating system:

  Windows Internet Explorer, Firefox: %AppData%\..\LocalLow\LastPass\ or %AppData%\..\Local Settings\Application Data\LastPass\

  Windows Google Chrome: %AppData%\..\Local\Google\Chrome\User Data\Default\databases\chrome-extension_hdokiejnpimakedhajhdlcegeplioahd_0\
  Windows Safari: %AppData%\..\Local\Apple Computer\Safari\Databases\safari-extension_com.lastpass.lpsafariextension-n24rep3bmn_0\

  OS X Safari, Firefox: ~/Library/Application Support/LastPass/
  OS X Firefox (non-binary): Firefox profile directory
  OS X Google Chrome: ~/Application Support/Google/Chrome/Default/databases/chrome-extension_hdokiejnpimakedhajhdlcegeplioahd_0/

  Linux Firefox: ~/.lastpass/
  Linux Google Chrome: ~/.config/google-chrome/Default/databases/chrome-extension_hdokiejnpimakedhajhdlcegeplioahd_0/

  Firefox on other platforms utilizes the Firefox profile directory.