How to show a featured image inside a post in only 1 single category

You can use in_cateogry function to check if the current post is in your special category. And if it is, you can then display the featured image. Something like:

if( in_category( 'YOUR CATEGORY ID OR SLUG', get_the_ID() ) ){
    the_post_thumbnail();
}

Obviously replace ‘YOUR CATEGORY ID OR SLUG’ with proper value.