Question

$\mathrm{The}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{expression}\tan 1°\times \tan 2°\times \tan 3°\times ...\times \tan 89°\mathrm{is}$

• A. \N
• B. 1
• C. 2
• D. 3

Answer 1

The correct option is B 1

Here, we can write the expression as
$\tan 1°\times \tan 2°\times ...\tan 89°\Rightarrow \tan 1°\times \tan 89°\times \tan 2°\times \tan 88°\times ...\times \tan 44°\times \tan 46°\times \tan 45°$
From the identity $\tan (90°-\theta )=\cot \theta$
we get $\tan 1°=\tan (90°-89°)=\cot 89°\tan 2°=\tan (90°-88°)=\cot 88°\mathrm{similarly}\tan 44°=\cot 46°$
Thus our expression becomes:
$\cot 89°\times \tan 89°\times \cot 88°\times \tan 88°\times ...\cot 46°\times \tan 46°\times \tan 45°$
Also $\tan \theta \times \cot \theta =1$
So we get $\tan 1°\times \tan 2°\times ...\tan 89°=1$