I also had this problem, i removed all the unused plugins and it was resolved

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I have exactly the same problem but the other way around: I can sign in with Firefox but not Safari.

Re previous answer: what plug ins?

you may check if all your plugins are up-to-date. To do this, go to the Mozilla Plugin Check site.

Once you're there, the site will check if all your plugins have the latest versions. If you see plugins in the list that have a yellow Update button or a red Update now button, please update these immediately.

To do so, please click each red or yellow button. Then you should see a site that allows you to download the latest version. Double-click the downloaded file to start the installation and follow the steps mentioned in the installation procedure.

I suspect a problem with cookies.

Make sure you allow cookies for Youtube. If you check Tools > Page Info > Permissions while on the site, you can make sure that Youtube is allowed to Set Cookies.

Second, you can clear any old cookies that might be corrupted or conflicting. In the Security tab of that same dialog, click the View Cookies button and remove each Youtube.com cookie.

Then try reloading the page. Any luck?

Sorry, don't know how to do these steps in Safari.