Why would we combine these Psalms?
1. The same refrain appears in 42:5, 11; 43:5
2. There are other similar lines- compare 42:9; 43:2
3. There is no heading to Psalm 43 in the Hebrew text

These are the first Psalms we have studied ascribed to someone other than David. The "sons of Korah" are mentioned in the heading of Psalms 42, 44-49, 84-85, 87-88. They are apparently descendants of Korah the Levite of Num. 16 who led a rebellion against Moses and Aaron and perished. His sons did not perish- Num. 26:10-11. They became temple musicians and gatekeepers- I Chron. 26.

Psalms 42 and 43 speak of despair
42:5, 11; 43:5 "Why are you in despair, O my soul?"
42:3 Tears were his food continually 
42:7 He feels as if he is drowning.
42:9; 43:2 he is continually mourning

These Psalms speak of hope
42:5, 11; 43:5 Hope in God
42:8 God gives His lovingkindness and songs in the night
43:3-5 He longs to be at God's holy hill

These passages describe an intense desire for God
42:1-2 His desire for God is compared to a thirsting and panting after water
42:4, 6 He remembers God and going with the group in worship of God
42:9 God is his rock
43:2 God is his refuge
43:4 God is his exceeding joy

This foreshadows Jesus
42:3 Tears are his food- Isaiah 53:3 He is a "man of sorrow"
43:3 In Jesus, God sent forth His light- John 8:12; 9:5- and His truth- John 14:6
They especially point to the cross
42:5, 11; 43:5 The word used in  the LXX for "despair" is used of Jesus  in Matt. 26:38; Mark 14:34
42:6 despair- A different Greek word is used in the LXX and the word is used of Jesus in John 12:27; 13:21
42:3, 10 They taunted Jesus saying, "Let God rescue Him" in Matt. 27:43
42:1-2 Thirst- Jesus was thirsty upon the cross- John 19:28- to give us the water of life- John 4:13-15; 6:35; 7:37-39.