# Prove this identity

Help me prove $\log\tan 1^0\cdot\log\tan 2^0\log\tan 3^0\cdot\cdot\cdot\log\tan 58^0\cdot\log\tan 59^0\cdot\log\tan 60^0=0$

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Hint: When is $\log x=0$? –  lhf Sep 21 '12 at 12:20

$\log\tan 1^0\cdot\log\tan 2^0\log\tan 3^0\cdot\cdot\cdot\log\tan43^0\cdot\log\tan44^0\cdot\log\tan45^0\cdot\cdot\cdot\log\tan 58^0\cdot\log\tan 59^0\cdot\log\tan 60^0=0$

Because $\log\tan45^0=\log 1=0$

$\log\tan 1^0\cdot\log\tan 2^0\log\tan 3^0\cdot\cdot\cdot\log\tan43^0\cdot\log\tan44^0\cdot0\cdot\cdot\cdot\log\tan 58^0\cdot\log\tan 59^0\cdot\log\tan 60^0=0$

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The question was tagged homework... –  lhf Sep 21 '12 at 12:20

When will a product of real numbers will be 0?

When will be $\log x=0$

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