Please reread Ernie50 explanation given above. You are given a tax credit for working in the state of New York. The property tax credit is better because your NY other state tax credit will offset all your NY income without needing to deduct property taxes. The property tax deduction is unnecessary to reduce your NJ tax liability (net of the NY credit).
Ernie50 has taken the time to work this out and he determined the credit was the best solution to take because by taking the deduction, the credit would be $3545 and $4182 without it.
Think of it this way. You can choose to take the $10,000 deduction but would you rather have $3545 in your pocket or $4182? The tax code can be tricky sometimes and confusing but it is in accordance to the way it is designed.
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