Why did Satan tempt Jesus in the desert?

There is no plausible reason for Satan to tempt Jesus in the wilderness, at least as described in Matthew 4:1-13 and Luke 4:1-11. Knowing that Jesus was the Son of God and knowing that Jesus would successfully and easily resist him, Satan would always have known that the effort was futile. Dale C. Allison Jr. ('How to Marginalize the Traditional Criteria of Authenticity', published in Handbook for the Study of the Historical Jesus, Volume 1, edited by Tom Holmén and Stanley E. Porter - page 14) does not believe Satan's motives need explaining:

Consider also Matt 4:1-11 = Luke 4:1-13 (Q), the temptation story. Most modern scholars have rightly judged this to be unhistorical, an haggadic fiction produced through reflection on scripture. Yet whoever composed it clearly did so in the knowledge that Jesus was (a) a miracle worker who (b) sometimes refused to give signs, (c) thought himself victorious over demonic forces, (d) was steeped in the scriptures, (e) had great faith in God, and (f) was a person of the Spirit. So what we seem to have in Q 4:1-13 is an illustration of the obvious fact that historical fiction can inform us about history. The story, which narrates events that probably never happened, nonetheless catches Jesus in several respects. Here the inauthentic incorporates the authentic.

Why did Satan tempt Jesus in the desert?

Because Satan hates Christ and wants him to fall.

What was Satan's motive in asking Jesus for a sign of His identity?

Because Jesus was going to save people by living a sinless life and dying for them. Satan didn't want that. He was tempting Jesus to fulfill a lust of using God's divine power to satisfy his hunger so that Jesus would fall. Satan could then make Jesus not able to save other on account that he was no longer sinless.

One of Satan's biggest tricks is to try and confuse you about who you are, and whose you are.

Satan will wait until you are at your weakest point in life and then come to you with questions about your life... your calling... and your ability to do for God the Father.

You must know that you have been given the power to become a son of God... Just as Jesus was his son in whom God was WELL PLEASED.

The Holy Spirit also not only Satan has a motive in the temptation of Christ:

" Then Jesus was led by the Spirit into the desert to be tempted by the Devil"( Matthew 4:1 NIV)

The motive of the Holy Spirit is to prove that the Son is God just as his Father is:

Temptation 1: If Jesus is really God's own Son, consubstantial with him, then he surely has the ability to turn those stones into bread.

( This is selfish and not the essence of life --Philippians 2:1-11)

Temptation 2: If Jesus is really God's eternally begotten Son, superior to angels, then he surely cannot be void of his servant's service, he cannot be hurt when he falls as it is written.

( This is not the divine will of the Trinity albeit possible and even good -- Mark 14:35)

Temptation 3: Satan commands Jesus to worship him and he will give his world to him.

(Satan does not understand the hypostatic union-- 1 Peter 1:12)

In every instance of temptation, Christ appealed wisely to the Scriptures for he himself is God's Wisdom ( Greek: sophia theou) --1 Cor. 1:24-25

I don't read that as Satan asking Jesus to prove his identity; rather, it reads as a temptation. "You have power. You're very hungry. Use your power to take care of your personal problems."

As Paul points out,

For we have not an high priest which cannot be touched with the feeling of our infirmities; but was in all points tempted like as we are, yet without sin. (Hebrews 4:15)

Satan absolutely did know who Jesus was, and why he was there. And he also knew that Jesus had free will and the ability to make his own choices. (In fact, we have one recorded instance where Jesus directly stated that his will was in opposition to the will of the Father, in the Garden of Gethsemane, (Matthew 26:38-44,) but he chose to do his Father's will and not his own.) And he knew that Jesus needed to be perfect to complete the work that he had come to accomplish.

So if he could find any way of tempting Jesus to step off the path, even once, it would have destroyed God's plan, which is exactly what Satan wanted. So Satan tempted him, trying to get him to use his power for physical desires, for fame, and for wealth and earthly power.