The value of $\frac{\tan55^\circ}{\cot35^\circ}+\cot1^\circ\cot2^\circ\cot3^\circ....\cot90^\circ,$ is:
A$-2$
B$2$
C$1$
D$0$
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C$1$
We have to find the value of the following expression
$\frac{\tan55^\circ}{\cot35^\circ}+\cot1^\circ\cot2^\circ\cot3^\circ....\cot90^\circ$
$=\frac{\tan55^\circ}{\cot35^\circ}+\cot1^\circ\cot2^\circ\cot3^\circ ....\cot90^\circ$
$=\frac{\tan(90^\circ-35^\circ)}{\cot35^\circ}+\cot(90^\circ-89^\circ)\cot(90^\circ-88^\circ)\\\ \ \ \ \cot(90^\circ-87^\circ) ....\cot87^\circ\cot88^\circ\cot89^\circ ....\cot90^\circ$
$=\frac{\cot35^\circ}{\cot35^\circ}+\tan89^\circ\tan88^\circ\tan87^\circ ....\cot87^\circ\cot88^\circ\cot89^\circ....\cot90^\circ$
$=1+1\times1\times1\ ....\times\ 0$
$=1$
As $\cot90^\circ=0$
Hence the correct option is $(c)$
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