You can do it using Nintex forms, by Rules Engine. Please refer below links.
Show/Hide fields in Nintex
If you don't want a Nintex form, You can add a content Editor/Script editor in the page, and capture the On Change event of the Drop down. So that you can Show/Hide any fields. In this case, you can get the any field values by its title. The DOM element title for all the fields is same as the "Title" (Display text) of the field. If the field is "Required", the field DOM element title will like "<"Required">". For example Title field DOM element will be a input field with the type of Text and the title property of the element will be "Title Required" . So you can get the value by using it.
For Ex:
My Field Name is "Other" and it is required and Type of DropDown. If i need to capture the onchange event of the field means i need to do like below.
$(select[title='Other Required']).change(function() {
if(this.val()=="")
{
//your code goes here
}
});
If that field is not required.
$(select[title='Other']).change(function() {
if(this.val()=="")
{
//your code goes here
}
});
Hope this helps you. Let me know if this helps.
Thanks.