Answer:

L.H.S
cot 4x (sin 5x + sin3x)
= cot 4x (2sin(5x+3x)/2 . cos (5x-3x)/2 )
= cot 4x (2 sin4x cosx)

= (cos4x / sin 4x).(2 sin4x cosx)
= 2cos4xcosx
R.H.S
cot x (sin 5x – sin3x)

= cot x (2cos(5x+3x)/2 . sin(5x-3x)/2)
= cot x (2 cos4x sinx)
= (2 cos4x sinx)
= 2cos4xcosx
L.H.S=R.H.S
Hence, proved.
Using the formula,

sinA + sinB = 2sin(A+B)/2 . cos(A-B)/2
sinA – sinB = 2cos(A+B)/2 . sin(A-B)/2