Is the site"hostgator"a trusted site for giving it my credit and if it not please tell me another trusted one

Yes, it is, and the security certificate is good through next March.

You can view the certificate info if you use Internet Explorer version 7: In Internet Explorer, you will see a “lock” icon in the Security Status bar. The Security Status bar is located on the right side of the Address bar.

Remember though that even though they have a valid security certificate, that does not mean that the business is actually legitimate. The security cert certifies only the data going back and forth.

The certificate that is used to encrypt the connection also contains information about the identity of the website owner or organization. You can click the lock to view the identity of the website.

I think most of web hosting sites can be trusted, along with hostgator, and I have using hostgator hosts for 2 year, it’s really a good hosting company, I love it’s price and support.
Besides here is a review of top 10 web hosting, if you want to buy web hosting, hope you take a look first.
http://review-pedia.com/index.php/Web-Hosting/Servers/Top-10-Web-Hosting.html