Drawing inferences from your partners card

You have led low and dummy has only low cards. Because the third player should play high, but cheaper of touching highest cards, you can draw huge inferences.

What happened
West led ♠ 3 vs 3 NT, and Trick One went ♠ 3, ♠ 2, ♠ J, ♠ Q. At Trick Two declarer led ♣ 9, and West won ♣ K. He then led ♠ 6. Declarer won ♠ 10 and drove out ♣ A. West persevered with ♠ 8, but declarer could win ♠ A and cash his minor-suit winners. Game made plus one.
What should have happened
Trick One reveals to West that declarer holds ♠ A (partner would have played ♠ A as third hand high if he held the card). It also reveals that declarer holds ♠ 10 - East would have played the cheaper ♠ 10 from ♠ J10. So when at Trick Two West wins ♣ K, he must not lead a second spade, rather try to put East on play for a spade through declarer’s known ♠ A10. Looking at dummy, it is clear for West to switch to hearts (the weakness), and he selects ♥ 9, as a high for hate lead. East wins ♥ A and reverts to ♠ 9 (top from two remaining). West beats declarer’s ♠ 10 with ♠ K, returns ♠ 6 to ♠ A, then wins the second club with ♣ A and cashes ♠ 8. Down one.
If you remember one thing...
Partner’s third hand play (high but cheaper of touching highest) can tell the leader a huge amount about the layout of the suit.