What is the purpose of using the word "coequal"

While the dictionary entries you cited are technically correct, I think they are missing the sense of the word, which I think is what really explains it. I work in programming, and deal with a similar distinction at work, where "=" "==" and "eq" all have nuances of the same basic meaning, 'equals'.

To say two things are equal implies actual, qualitative similarity. Equal means two things ARE the same, at least with regards to the quality you're talking about. Being coequal, rather, is a more general statement about the relative standing of the things being considered.

All people have equal rights. But you and your boss are not coequals, there is a difference in standing.

From your example, if you said that all three branches of government were equals, hits the ear funny, equals with regards to what? You can say they have equal power. You can say they have equal standing. But they are coequal.

To back this up with a reference, here you go:

https://wikidiff.com/equal/coequal

This cites a similar difference as I point out above. Equal points to actual equality, while coequal says the things are "equal to each other in size, rank or position."

In seeking to identify the original perceived need for the word coequal in English, I decided to focus on the context in which it tended to appear in its early days of written usage. That context was primarily religious discussion.

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Early print instances of 'coequal' or 'coequall'

Searches for coequal and coequall in the Early English Books Online database return fourteen matches from before 1550. Of these, thirteen involve religious themes, and ten specifically involve the nature of the Trinity. Following are the fourteen matches.

From "Here Begynneth a Treatyse of this Galaunt with the Maryage of the Bosse of Byllyngesgate. vnto London Stone (1521?):

Therfore let my wyfe and me alone. / For by my study and wakynge many a nyght. / I knowe by the sterres / that shone by the moone. / That fayre Bosse / hooly was in my syght / And that to my nature / she sholde be coequall. / And remayne as my fere / euer in my syght. / By the purueyaunce / of the goddes Imperyall / To my comforte shynynge as the sterres bryght

According to the Hidden London website, "The Boss of Billingsgate was some sort of post, probably a drinking fountain, that stood outside the fish market. One source suggests that the boss bore an image of Belin, another that it was shaped like a pot-bellied man. It was a tradition that fish porters would ask passers-by to kiss the boss and, if they refused, they were bumped against it. It was the subject of a comic ballad, entitled ‘The Marriage of London Stone and the Boss of Billingsgate’ (1521), and a play by John Day and others (1603), which has not survived." Wikipedia has a lengthy article about "London Stone," although it asserts that "The earliest reference to the Stone is usually said to be that noted by John Stow in his Survey of London (1598)"—which is roughly 77 years after the ballad quoted above.

From A. C. Mery Talys (1526?):

In a wyllage in warwyk shyre there was a parysh prest all though he wer no great clarke nor grad[v]at of the vnynersyte / yet he [p]rechid to hys paryshons vppon a s[vn]aday / [de]claryng to th[em] xii. artycles of the Crede. Shewyng them that [t]he furst 〈◊〉 was to be[lieve] god the fader almyghty maker of heuen and erth. 〈◊〉 T[o] bele[ive]〈◊〉 cryste hys onely son our lorde coequal wyth the fa[ther] in all [th]yngys 〈…〉 to the deyte.

From Catholic Church, "Here After Foloweth the Prymer in Englysshe Sette Out Alonge, After the Vvse of Sarum" (1538):

[T]he Father is made of none: neyther created nor gotten. / [T]he Sone is frō the Father alone: neyther ma∣de ne create / but gotten. / [T]he holy Ghost is frō the Father / and the Sone / neyther made created / nor gotten but procedynge. / [A]nd so there is but one Father / not thre Fathers / one Sonne / not thre Sonnes / one holy Ghoste / not thre holy Ghostes. / [A]nd in this Trinyte / there is none before or after another / nothynge more or lesse: but all the thre persons be coeterne / and coequall to them selte. / [S]o that by alwayes as now it hath ben aboue sayd the Trinyte in vnite / and the vnite in Tri∣nyte may be worsshypped. / [H]e therfore that wyll be saued / let hym vnderstande thus of the Trinyte.

From "The Seconde Canonicall Epistle of Peter the Apostle, in The Newe Testamente Both Latine and Englyshe Ech Correspondent to the Other After the Vulgare Texte, Communely Called S. Ieroms (1538):

Simon Peter the seruaūt and Apostle of Christ Iesus, vnto them that haue optayned coequall faythe wyth vs in ye ryghteousnesse of oure God, and the saueoure Iesus Christ ⋆ Grace be vnto you and peace be fulfylled in the knowlege of God, & Christ Iesu our LORD.

From Church of England, The Manuall of Prayers, or the Prymer in Englyshe set out at Lengthe, whose Contentes the Reader by the Prologe Next after the Kalendar, Shal Sone Perceaue and There in Shal Se Brefly the Order of the Whole Boke (1539):

The father is made of none, neyther created, nor gotten. The sonne is from the father alone, neyther made nor created, but gotten. The holy goost is from the father, and the sonne, neyther made, created nor gotten, but procedyng. And so is there but one father, not thre fathers: one sonne, not thre sonnes, one holy goost, not thre holy goostes. And in this Trinite, there is none before or after another, nothyng more or lesse, but al the thre personnes be coeterne, and coequall to themselfe. So that by all wayes as nowe it hath ben aboue sayde, the Trinite in vnite, and the vnite in trinite maye be worshypped.

From Heir Beginnis the Hystory and Croniklis of Scotland (1540?):

During this tyme, the cristin faith apperit to rise in gret dignite. Nochtheles the cursit heretik Arrius began to poyson it with vennymus doctrine. Saying Crist was nocht the verray sonne of god, coequall and coeternall to the fader bot different fra hym in substance. Efter degeist consultatioun all his opinionis war condampnit in counsall of Nicia with mony othir vane errouris, quhilkis I wyl not rehers at this tyme.

From The Epistles and Gospelles with a Brief Postil vpon the Same from after Easter tyll Aduent (1542):

Good people we be sure by scripture, that there is but one God. For it sayth. Herken o Israel, thy god is one. But forasmuch as the scripture doth attribute godheade and diuine essencie to thre: therfore the fathers haue founde out the worde (person) for the auoydynge of many errours. And hereof for discernynge the sayd persons is the name of Trinitie come into the church, wherby we sygnifye not .iij. vnegall persons, but thre persons coequal of one in diuisible substaunce and essencie.

From Aergur Kelton, A Commendacyun of Welshmen (1546):

His [Eneas's] marciall warres / Surmounting the starres / Made hym celestiall / After pagan guyse / In suche maner wyse / With ther goddes to be coequal

In Libie land / His temple did stand / Freate with gold pearle & stone / By deuine Oracle / His tabernacle / Made ther ful long agone

From John Bale, A Tragedye or Enterlude Manyfestyng the Chefe Promyses of God unto Man by All Ages in the Olde Lawe from the Fall of Adam to the Incarnacyon of the Lorde Iesus Christ (1547?):

In the begynnynge, before the heauens were create, / In me and of me, was my sonne sēmpyternall. / With the holy Ghost, in one degre or estate, / Of the hygh Godhed, to me the father coequall. / 〈…〉hys my sonne was, with me one God essencyall, / [Wit]hout separacyon, at any tyme from me. / 〈…〉 God he is, of equall dignyte. / 〈…〉 the begynnynge, my sonne hath euer be, / 〈…〉s father, in one essencyall beynge.

From a 1548 translation of Martin Luther, The Chiefe and Pryncypall Articles of the Christen Faythe to Holde Againste the Pope and Al Papists, and the Gates of Hell:

So the Father is the Lord, the Sonne the Lord, the holy ghoost the Lorde. And yet be they not three Lordes, but one Lorde. For as we be compelled by the christian veritie, to cōfesse separately euery one person to be God and Lorde. So are we prohibite by the Catholyke relygyon of Christes faythe to saye: that there be three Goddes or three Lordes. The father is made of none, neyther crated, nor gotten. The sonne is from the father alone neyther made nor created, but gotten. The holy Ghost is from the father and the sonne, neyther made, created, nor gotten, but procedynge. And so is there but one father, not three fathers, one sonne, not three sonnes, one holy Ghoost, not three holy ghoostes. And in thys trinite, there is none before or after another, nothyng more or lesse. But al the three persōs be coeterne and coequal to them selfe.

From The Booke of the Common Prayer and Administracion of the Sacramentes, and Other Rites and Ceremonies of the Churche: After the Vse of the Churche of England (1549):

The father is made of none: neither created nor begottē. The sonne is of the father alone: not made nor created, but begotten. The holy gost is of the father and of the sonne: neyther made nor created, nor begotten, but proceding. So there is one father, not three fathers: one sonne, not three sonnes: one holy gost, not three holy gostes. And in this trinitie, none is afore nor after other: none is greater nor lesse then other. But the whole three persons: be coeternall together and coequall. So that in all thinges, as it is afore sayed: the vnitie in trinitie, and the trinitie in vnitie, is to be wurshipped. He therfore that will be saued: must thus thinke of the trinitie.

From a 1549 translation of Desiderius Erasmus, The Praise of Folie:

And yet shall one of those shamefast, maidenly men not sticke than to d[i]splaie his pecockes fethers, and rowse hym selfe, whyles suche shameles flaterers dooe goe about to make him, being a man lesse worth thā naught, coequall yet vnto the Gods, in blasonnyng hym for a paragonne, and absolute example of all maner vertues, from which he knoweth hym selfe to be as farre wyde, as from hence to the man in the moone.

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They are ministers of Christ, and so am I also, to the ende yet he shuld not be holden for a vainglorious vaunter in that he made him selfe coequall with the other, he added as by correction, I more than other, signifiyng therby, how not onely he was matche to the other apostles, but somewhat also their superiour: whiche althoughe he woulde shoulde be taken for verie trueth, yet lest the arrogant auowyng therof might partly haue offended mens •ares, he did first shelde the same with the pretexte of Folie, (saiyng) I speake it as the vnwyser, because he wist what priuilege fooles haue to speake trouthe without offence.

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And this was his exposicion. I speake it as the more vnwise, that is to saie (quod he) in case I seeme vnwise vnto you because I doe coequall my selfe vnto the false apostles, than more vnwise will you coumpt me, in auauncyng my selfe afore theim.

From John Proctor, The Fal of the Late Arrian (1549):

And hereunto all the olde fathers agre: and of our late doctors I knowe none that dissent in this poynt: so that it is euydente that ye haue soked this out of your owne brain only, as to contend that Christ is not god, or of the saine nature wt the father, because Paule sayeth Christe is Gods, or of God• Vpon which wordes Bullinger writeth thus: Christ is gods, that is to saye (sayeth he) Christ is coequall with the Father, and vnto him, the Churche is subiect, not as to man, but as to the lyuynge God.

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If ye can declare any to be equall with hym [Jesus] in these poyntes, then wyll I graunte with you that he hath mo brothers: & euery of them true God. For none but God can saye as he hath sayde: & to none but to God can it be sayde whiche ye haue hearde. In meane tyme I beleue him to be brotherles, and confesse him to be thonly begot∣ten sonne of God, consubstanciall & coequall to ye father in al power, honor, and glorye.

And from "Whippet You Priests" (1549?):

O Lorde thou dyddest thy fathers wrath pacify / ♣ •Obedient thou wast vnto a shamefull death / For mans lyfe, thou suffredest patiently / Thou yeldest the gost, as the scripture sayth / And rose from death, to lyfe the thyrde daye / And sittest in heauen, with great power & maiestie / ♣ ♣ Coequall with the father, thys is no haye / Makyng intercession, for vs synners perpetuallye.

Also from Thys Booke Is Called the Treasure of Gladnesse and Semeth by the Copy, (Beeing a Verye Litle Manuel, and Written in Velam) to Be Made Aboue CC. Yeres Past at the Least (1563):

My Testament.

This I make my Testament, and laste will: in the name of the eternall liuynge God: the father, the sonne & the holy ghost: in whose name I was baptized. In whome onely I hope. and beleue to be saued. Amen. First I bequethe my soule into thy hands: O god. father: sonne, & holy Ghost. Thou hast first made me and thou hast gyuen thy sonne to becom man: and died for my sinnes & for the sinnes of the people, O father for thy sonnes sake haue mercy vpon me. O lorde Iesu Christ, thou sonne of god, thou haste boughte me with thy pre∣cious bloud. By one oblacion sufficiently, for all that beleue in thee. O Christ, god and man, which art in heauen, haue m[e]rcy vpon me, & be thou my mercifull mediator for me vnto thy father, that I m[a]y be saued. O holy Ghost god, coequal with the father and the sōne, haue mecry vpō me, worke thy deuine power in me, through thy gracious inspiracion, draw me vnto Iesu Christ, yt I may find fauor & be saued. Amē

I have included this last instance among the pre-1550 texts because it has some claim to being quite old. In describing a 1615 edition of this work, a PDF posting on the "[2016] New York Book Fair Highlights" from the Marlborough Rare Books website offers the following remarks in connection with a 1615 edition of The Treasure of Gladnesse:

This Elizabethan best-seller survives in only a handful of copies. NSTC enumerates twelve separate editions between 1563 and 1601 ...

Clearly The Treasure of Gladnesse was derived from a mediaeval text, translated to English, and given a light Protestant makeover that was acceptable to Puritan sensibilities. In this late, if not last edition, not only is a more conventional spelling now introduced, and side notes to the scriptures dispensed with, but variants and changes, more or less substantial, were made to the text which better reflected the course of religious enthusiasm palpitating through the Jacobean period.

After 1615 works based on a religious text over two centuries old and written on vellum may have looked like something almost tainted and too closely allied to the past to be safely printed. The Treasure, even though it had been a valuable and constant best-seller, probably quickly fell out of favour with publishers and public alike and accounts for the inadequate number of surviving copies.

I have no idea whether the word coequal appeared in English in the source material for this text, but it seems likely that if the passage did appear in a vellum text that was at least 200 years old in 1563, it was written in Latin.

The odd example out among the preceding fifteen early instances of coequal/coequall is the earliest one—the comic song from 1521 involving the assertion by the husband London Stone that his wife, the Bosse of Byllingesgate, should be "coequall" to his nature. I have no insight into the writer's choice of coequall in this instance.

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The usefulness of 'coequal' as a word distinct from 'equal'

The locus of early usage of coequal appears to be the Athanasian creed, which pairs coequal and coeternal as essential attributes of the Father, the Son, and the Holy Spirit.

It is undoubtedly true that a disproportionate number of early texts in the EEBO database focus religious matters—but the specificity of the use of coequal in these tracts as a a way of expressing the relative status of the Father, the Son, and the Holy Spirit seems to me to be central to the word's injection into English discourse as well as to its continuation there, at least over the next two centuries. In Christian doctrine—both Catholic and Protestant—the word coequal served the critical purpose of emphasizing the unitary equality of the three aspects of the Trinity, as opposed to their being in any sense separate but equal.

A Christian polemicist of this period might accept the assertion that Zeus, Jupiter, and Odin—false gods though they might be—were equal, but it wouldn't do to assert that the Father, Son, and Holy Spirit were, because that familiar term failed to address the critical "withness" of their equality. The word coequal thus expressed an essential element of equality that equal did not, just as the word coeternal indicated an interconnection that eternal did not. In simple terms, a thing may be equal to something else, but coequal with it.

The earliest dictionary definition of coequal that I've been able to find is from John Bullokar, An English Expositor: Teaching the Interpretation of the Hardest Words Used in Our Language (1616):

Coequall. Equal in degree with another.

Notably, Bullokar does not include an entry for equal, presumably because he didn't consider it to be among the "hardest words" requiring interpretation.

In subsequent usage, the centrality of religious applications of coequal has fallen away. In the United States, in particular, it almost a cliché to speak of the executive, legislative, and judicial branches of the federal government as "coequal branches." In their essential meaning, I see little distinction between "equal branches" and "coequal branches"—but there is still at least a hint of implied cooperative status in the latter expression that is not necessary or inherent in the former.