The value of $\tan1^\circ\tan2^\circ\tan3^\circ.....\tan89^\circ$ is:
A$1$
B$-1$
C$0$
D
None of these
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A$1$
Here we have to find: $\tan1^\circ\tan2^\circ\tan3^\circ.....\tan89^\circ$
$\tan1^\circ\tan2^\circ\tan3^\circ.....\tan89^\circ$
$=\tan(90^\circ-89^\circ)\tan(90^\circ-88^\circ)\tan(90^\circ-87^\circ)\ \\ \ \ \ \ ...\tan87^\circ\tan88^\circ\tan89^\circ$
$=\cot89^\circ\cot88^\circ\cot87^\circ...\tan87^\circ\tan88^\circ\tan89^\circ$
$=(\cot89^\circ-\tan89^\circ)(\cot88^\circ\tan88^\circ) \\ \ \ \ \ (\cot87^\circ\tan87^\circ)...(\cot44^\circ\tan44^\circ)\tan45^\circ$
$=1\times1\times1...1\times1$ $[\text{since}\cot\theta\tan\theta=1]$
$=1$
Hence the correct option is $(a)$
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