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Simulate backspace button JS

I want to create a custom backspace button with the same logic as “backspace” button on a keyboard. I use the following code: 

function backSpace()
{
    var e = jQuery.Event("keyup");
    e.which = 8; // # Some key code value
    e.keyCode = 8;
    $("mainDiv").trigger(e);
}
<!DOCTYPE html>
<html>
    <head>
        <link rel="stylesheet" href="styles.css" type="text/css" media="screen" />
        <script src="formulas.js"></script>
        <script src="http://code.jquery.com/jquery-1.11.3.js"></script>
        <meta name="viewport" content="width=device-width initial-scale=1.0 maximum-scale=1.0 user-scalable=no"/>
    </head>

    <button onclick="backSpace()">backSpace</button>
    <body id="main" spellcheck="false">
        <div id = "mainDiv" contenteditable="true"></div>
    </body>
</html>

But it doesn’t work. I don’t understand what I’m doing wrong. I spend a lot of time for this problem, but I haven’t solved it yet. Help me, please.

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Answer

Addition to @CosLu’s Answer

JSBin Here also

function backSpace() {
  p = document.getElementById("mainDiv");
  c = getCaretPosition(p);
  console.log(getCaretPosition(p));
  str = $("#mainDiv").html();
  if (c > 0 && c <= str.length) {
    $("#mainDiv").focus().html(str.substring(0, c - 1) + str.substring(c, str.length));

    p.focus();
    var textNode = p.firstChild;
    var caret = c - 1; 
    var range = document.createRange();
    range.setStart(textNode, caret);
    range.setEnd(textNode, caret);
    var sel = window.getSelection();
    sel.removeAllRanges();
    sel.addRange(range);
  }
}

$.fn.setCursorPosition = function(pos) {
  this.each(function(index, elem) {
    if (elem.setSelectionRange) {
      elem.setSelectionRange(pos, pos);
    } else if (elem.createTextRange) {
      var range = elem.createTextRange();
      range.collapse(true);
      range.moveEnd('character', pos);
      range.moveStart('character', pos);
      range.select();
    }
  });
  return this;
};

function getCaretPosition(editableDiv) {
  var caretPos = 0,
    sel, range;
  if (window.getSelection) {
    sel = window.getSelection();
    if (sel.rangeCount) {
      range = sel.getRangeAt(0);
      if (range.commonAncestorContainer.parentNode == editableDiv) {
        caretPos = range.endOffset;
      }
    }
  } else if (document.selection && document.selection.createRange) {
    range = document.selection.createRange();
    if (range.parentElement() == editableDiv) {
      var tempEl = document.createElement("span");
      editableDiv.insertBefore(tempEl, editableDiv.firstChild);
      var tempRange = range.duplicate();
      tempRange.moveToElementText(tempEl);
      tempRange.setEndPoint("EndToEnd", range);
      caretPos = tempRange.text.length;
    }
  }
  return caretPos;
}
<!DOCTYPE html>
<html>

<head>
  <meta charset="utf-8">
  <title>JS Bin</title>
  <script src="http://code.jquery.com/jquery-1.11.3.js"></script>
</head>

<body>
  <button onclick="backSpace()">backSpace</button>
  <div id="mainDiv" contenteditable="true">aahahtext</div>
</body>

</html>
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