Basically, you want to store the string <mailto:example@hotmail.com> "example@hotmail.com" into the $to_email variable (you're having a similar problem with the $from_email variable).
In order to do this, you need to surround the string in quotes. Since you use double quotes inside the string, I would suggest you simply use single quotes, like so:
$to_email = '<mailto:example@hotmail.com> "example@hotmail.com"';
For more information, read in the manual about what a [man]string[/man] is.
Also, I've removed the actual e-mail addresses from your post - don't want to collect spam, do you? 😉
EDIT: OI! Bpat! Lay off, I was here first! :p
EDIT2: Also, unless you have had problems doing this in the past (meaning you have a poor-quality MTA installed on your server), the line endings after all of the headers in the mail() command should be CRLF's, or "\r\n", as described on the manual page for the mail() function: [man]function.mail[/man].