Where does it show in the Bible that Jesus is Michael the archangel?

Scriptural evidence indicates that the name Michael applied to God's Son before he left heaven to become Jesus Christ AND ALSO after his return.

Archangel means "chief angel," or "principal angel" indicating that there is only one. Michael is the only one said to be "the archangel". Jude 9 says: "Michael THE archangel had a difference with the Devil and was disputing about Moses' body."

To illustrate that Michael is another name for Jesus, at 1 Thessalonians 4:16 the voice of the resurrected Lord Jesus Christ is described as being that of an "archangel", suggesting that he is, in fact, himself the archangel. If the designation "archangel" applied, not to Jesus Christ, but to other angels, then the reference to "an archangel's voice" would not be appropriate. In that case it would be describing a voice of lesser authority than that of the Son of God.

Also, Daniel, after making the first reference to Michael (Da 10:13), recorded a prophecy reaching down to "the time of the end" (Da 11:40) and then stated: "And during that time Michael will stand up, the great prince who is standing in behalf of the sons of [Daniel's] people." (Da 12:1) Michael's `standing up' was to be associated with "a time of distress such as has not been made to occur since there came to be a nation until that time." (Da 12:1) In Daniel's prophecy, `standing up' frequently refers to the action of a king, either taking up his royal power or acting effectively in his capacity as king. (Da 11:2-4, 7, 16b, 20, 21) This supports the conclusion that Michael is Jesus Christ, since Jesus is Jehovah's appointed King, commissioned to destroy all the nations at Har–Magedon. (Rev. 11:15; 16:14-16)

Also, the book of Revelation (12:7, 10, 12) specifically mentions Michael in connection with the establishment of God's Kingdom and links this event with trouble for the earth. Since the Son of God (Jesus) is to fight the nations, it is only reasonable that he was the one (also known as Michael) who with his angels earlier battled against the superhuman dragon, Satan the Devil, and his angels.

For more, see
Who Is Michael the Archangel?