I had a friend say that Hecate and Ishtar are actually the same goddess. In your research do you find this to be false or have you ever heard of this comparison? If so how would you prove this person wrong?
First: I apologize, I don’t know when this came through. I did not receive a notification from Tumblr.
To your question, the TLDR upfront: Academically, I have no evidence to suggest they are the same or even all that similar. I have not heard of this claim before this question. I don't think "proving them wrong" is fruitful.
Hekate is often compared to dark goddesses (such as Morana, Lilith, Kali, etc) but that doesn’t mean they are the same being. “Equivalence” in this context means “similar” or “performing the same function” - it does not mean they are the same entity. This is a distinction that often is often lost when we talk about the religion of ancient cultures - and especially those cultures that interacted with and influenced each other. Religion is fluid, not static, and the stories and characterization change over time. Syncretism of similar deities (such as the Greek and Egyptian Isis and Dionysus) can occur between cultures that have a great deal of activity with each other over a long period of time - to which we have ample evidence thereof. Moreover, human belief tends to "rhyme"; we see and generate similar themes in response to our environment, the way we live, and who we interact with. This creates such phenomenon as multiple cultures with chthonic earth mother-goddesses (and their later displacement by sky father-gods of invading peoples). It's not unreasonable for us to create or recognize similar patterns in different places. However, just as we wouldn’t claim that the Greek Demeter is the same as the Nordic Sif because they both hold domain over wheat, so too should we resist consolidating every goddess with a similar aspect. It diminishes and strips away Her cultural and historical context in the process. This is monolithic thinking.
But then, I would question what similarity is being drawn between the two in question, exactly? Ishtar/Inanna (as they became syncretized during the reign of Sargon of Akkad) was an Akkadian (modern day Iraq) sex-cult goddess of war. Characteristically she has more in common with the Greek Aphrodite (though I would not consider them the same). Her worship can be attested between roughly 3000 BCE to 1600 BCE. The earliest record of Hekate is in Hesiod's Theogony in C 8 BCE, and up until at least the post-Christian Chaldean Oracles (C 2 or 3 CE) and her influence spanned primarily Greece, Rome, and Anatolia (modern day Turkey). There isn’t a great deal of overlap, and no suggestion of syncretism between Hekate and Ishtar.
For my small part of research, the claim seems to stem from a tenuous echo of Inanna's descent into the underworld to retrieve her lover, Tammuz, with Hekate’s escorting Persephone to and from Hades. This, and Hekate’s association with the moon and childbirth (not sexual love/fertility; "fertility" being a loaded term flung haphazardly at just about any female deity), just as Diane/Artemis has an association with the moon and childbirth (both virgin goddesses Hecate and Diane appearing on the Pergamon tablet) is, again, tenuously compared to Inanna/Ishtar's aspect of the moon and "fertility" (sexual love and the fecundity of dates, specifically). However, Inanna/Ishtar is more extensively associated with the planet Venus (perceived then as a star) than with the moon - she formed a triad with Shamash (the sun god) and Sin (the moon god). Her symbol is an eight pointed star. Ishtar/Inanna is primarily a goddess residing in the heavens (what the Greeks would call ouranic) while Hekate is most often associated with earth/underworld (chthonic). I have found no significant similarities. Essentially: what is their claim, exactly, and what is their evidence to support it?
Finally, are they arguing this from a position of intuition/faith, or from a position of academic study? If their claim is academic they must have proof thereof and have scrutinized their sources. But it is utterly fruitless to argue that there is no, or little, or controversial, evidence to support a claim if they feel it is spiritually correct. Intuitively, I associate Hekate with Baba Yaga. There is no historical evidence linking them together, it simply seems right to me. What is intuitively true for me may not be spiritually true for someone else and visa versa; arguing the point will only inspire zeal and tension. And, ultimately, question whether or not zeal is what's motivating you to "prove them wrong." Are you reacting from a place of academic rigor (in which case you should be prepared to have that challenged and altered to new/better evidence) or are you reacting from a place of differing religious belief? Which scenario makes it appropriate for you to “debunk” your friend? And, if it’s not really the belief but their attitude that is causing rancor, evaluate that.