Q:

Why did Satan tempt Jesus in the desert? 

In Matthew 4:1-13 and Luke 4:1-11, Jesus was in the desert for 40 days and tempted by Satan.  Of particular interest is the first temptation. 

Luke 4:3-4 (NLT) 
3 Then the devil said to him, “If you are the Son of God, tell this stone to become a loaf of bread.” 
4 But Jesus told him, “No!  The Scriptures say, ‘People do not live by bread alone.’”    

Satan full well knew who Jesus was!  There couldn't have been any doubt in Satan's mind as to Jesus' identity, as opposed to the Pharisees and Sadducees, who simply refused to believe.  No, Satan isn't really asking Jesus to prove Himself.  Rather, he has some other motive...
What was Satan's motive in asking Jesus for a sign of His identity? 
I consider myself non-denominational, but I grew up Wesleyan, so the closer answers are to that, the better.  I won't turn away Catholic or Orthodox answers though. 

A:

The recorded three temptations of Christ mirror the steps Satan used to bring sin into the world in the first place:

If you are the Son of God ...

vs Genesis 3

Did God actually say ...

Also, like with the first sin, it is related to food. 
Second:

If you are the Son of God, jump off ...

vs 

You will not surely die

God didn't mean what He said ("I even quoted a passage to prove it!") 
And lastly:

if you will worship me ...

vs

you will be like God ...

Questioning God's authority. 
Satan is questioning whether Jesus was who He said He was (and whom His Father had just attested to at the baptism) - He was trying to get Jesus to fail in His earthly life so He couldn't be our perfect sacrificial substitute. 

A:

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.