Question : The value of $\tan1° \tan2° \tan3°...\tan89°$ is:
Option 1: $1$
Option 2: $0$
Option 3: $\sqrt{3}$
Option 4: $\frac{1}{\sqrt{3}}$
Correct Answer: $1$
Solution : $\tan1° \tan2° \tan3°...\tan89°$ $= \tan1° \tan2° \tan3°...\tan45°...\tan(90-3)°\tan(90-2)°\tan(90-1)°$ $=\tan1° \tan2° \tan3°...\tan45°...\cot3°\cot2°\cot1°$ As we know, $\tan\theta=\frac{1}{\cot\theta}$ So, it will cancel out all the terms except $\tan45°$. $=\tan45°$ $= 1$ Hence, the correct answer is $1$.
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