Check if a JavaScript variable exists and checking its value
First off, your code really breaks down to this logic and I find it helpful to write it out the longer version to full understand the logic flow before trying to shorten it:

var header_cookie, cookie_check, city_profile, state_profile;
if (typeof header_check !== "undefined") {
    header_cookie = 'user_profile';
    cookie_check = true;
    city_profile = 'userCity';
    state_profile = 'userState';
} else {
    header_cookie = 'admin_profile';
    cookie_check = false;
    city_profile = 'city';
    state_profile = 'state;
}


FYI, I also find this a LOT easier to follow what's actually happening than the code you have.  It also saves several comparisons on the cookie_check value.



There are some ways to shorten this, but it's not entirely clear that any are "better" where the definition of better includes readability by someone who has never seen this code before, but you can decide what you think of that issue for the alternatives:

Since you really only have two states, you could predefine each state and then just pick which one to use and access the properties off a single state object:

var userState = {
    header_cookie: 'user_profile', city_profile: 'userCity', state_profile: 'userState';
};
var adminState = {
    header_cookie: 'admin_profile', city_profile: 'city', state_profile: 'state';
};
var state = typeof header_check !== "undefined" ? userState: adminState;


Done this way, you'd access state.header_cookie, state.city_profile and state.state_profile rather than your standalone variables.



Or, if you wanted to keep the individual variables, you could do this:

var states = {
    header_cookie: ['user_profile', 'admin_profile'], 
    city_profile: ['userCity', 'city'],
    state_profile: 'userState', 'state'];
};

var stateIndex = typeof header_check !== "undefined" ? 0 : 1;
var header_cookie = states.header_cookie[stateIndex];
var city_profile = states.city_profile[stateIndex];
var state_profile = state.state_profile[stateIndex];