Discover that leaders sometimes use similar slogans in public (4,3)
I believe the answer is:
suss out
'discover' is the definition.
(I've seen this before)
'leaders sometimes use similar slogans in public' is the wordplay.
'leaders' suggests taking the first letters.
'in public' becomes 'out' (I've seen this before).
The first letters of 'sometimes use similar slogans' is 'suss'.
'suss'+'out'='SUSS OUT'
'that' acts as a link.
(Other definitions for suss out that I've seen before include "Discover" , "Check into" , "Examine to determine accuracy" , "Grasp" , "Solve" .)