Consider L.H.S.

= cot 12° cot 38° cot 52° cot 60° cot 78°

= (cot 12° cot 78°) (cot 38° cot 52°) cot 60°

= (cot 12° cot (90-12)°) (cot 38° cot (90-38)°) cot 60°

= (cot 12° tan 12°) (cot 38° tan 38°) cot 60°

= 1 × 1 × (1/√3)

= 1/√3 = R.H.S.

Hence, proved.