The Gemara in Berachos 12b records a Halacha that during the Asseres Yemei Teshuva one should replace “ha’el hakadosh” with “hamelech hakadosh” and “melech ohev tzedakah umishpat” with “hamelech hamishpat” during shemoneh esre. Over the years a lengthy addition was made to the berachah of “hamelech hakadosh” during Rosh HaShanah and Yom Kippur. However, even before the addition was made, the change in the wording is not easy to understand. The words “ha’el hakadosh” describe G-d as a transcendental force, one who is separated from the rest of the world. The Rav zt”l famously translated “kadosh, kadosh, kadosh” as “transcendental, transcendental, transcendental” (based on the comment of Rashi at the beginning of Parashas Kedoshim that “kadosh” means separate or distinguished). G-d is separate from the world, He is external to the world. His power is so overwhelming that He cannot even be related to. After thanking Him for helping our people through the ages and proclaiming how His goodness is made possibly by His total control over nature, we encounter His awesome, overwhelming, and above all inscrutable, power. That is the G-d of “ha’el hakadosh”. After that, the davening is essentially an attempt to bridge that enormous gap: to strengthen and add texture to our relationship with G-d. So on a weekday we ask Him to give us the knowledge to address Him properly, to repent before Him and be worthy of His presence. On Shabbat, we appeal to Him through the holiness of the day. During chagim we appeal to Him through our own chosenness. If He has chosen us to be His people, He must now provide us with the ability to address Him and, so to speak, to approach His holy throne. Seemingly, this is the subtext on Rosh HaShanah as well (for on Rosh HaShanah and Yom Kippur the middle beracha also starts with those words which speak of chosennes, “ata bechartanu”). However, on Rosh HaShanah and Yom Kippur we find one thing which is very different. On those days, instead of saying “ha’el hakadosh”, we say “hamelech hakadosh.” If so, the following text, “Ata bechartanu”, is not a way of connecting with G-d, it is a way of emphasizing an already existing connection. You are our holy king (as opposed to the inscrutable and transcendental force which controls the world), we say to Him, and You chose us to be Your people.
Let us now consider the additions to the third beracha on Rosh HaShanah and Yom Kippur. The beracha starts as it does on any other day: “You are holy (transcendental) and Your name is holy (transcendental) and the holy ones (Israel and the angels, those who are different) will every day praise You.” However, the beracha then goes on an excursis: “and for this reason all the nations will fear You and they will all bow down to You.” While the progression within most of the addition is easily understood (first the entire world will fear G-d and acknowledge His mastery over it, then they will give honor to G-=d’s chosen people, then the righteous will rejoice and G-d will return to the seat of His kingship on earth and restore the divine presence to Jerusalem) but the place of this addition in a beracha about the transcendental nature of G-d is not obvious. In fact, it seems to contradict that idea.
Towards the end of this addition to the third beracha of davening we find a proof text for the predictions our prayers make and the hope that they express: “May Hashem rule forever, your god oh Zion, for all generations Hallelujah!” This passuk seems to be the best possible proof text to site when talking about G-d’s kingship and so we would have thought that we would then naturally progress to the end of the beracha: “Blessed are you the holy (transcendental) king.” However, that isn’t what happens. The beracha has one more thing to add: “You are holy and Your name is fearful and there is no other god than You.” The dichotomy stairs us in the face, an a solution is provided: “You are holy (transcendental)” and so can’t relate to You, not really. And yet, “Your name is fearful”, we might not be able to relate to You or comprehend You, we might not be able to approach Your holy throne, and yet we can interact with your name. Your name carries an image and connotations. Your name provides the way that we can build our relationship with You. And what characterizes that relationship? “And Hashem the Lord of Hosts will be elevated in judgement and the holy (transcendental) G-d will be sanctified by justice.” G-d’s name provides us with our relationship with Him and that relationship is characterized most obviously by the fact that He judges us for our actions. Through that judgement He can be our “holy king.”