L.H.S

cot 4x (sin 5x + sin3x)

= cot 4x (2)

= cot 4x (2 sin4x cosx)

= (2 sin4x cosx)

= 2cos4xcosx

R.H.S

cot x (sin 5x - sin3x)

= cot x (2)

= cot x (2 cos4x sinx)

= (2 cos4x sinx)

= 2cos4xcosx

L.H.S=R.H.S

Hence, proved.

Using the formula,

sinA + sinB = 2sin cos

sinA - sinB = 2cos sin