Question

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

• A. $1$
• B. $2$
• C. $4$
• D. $0$

Answer 1

The correct option is A $1$

$\tan 1°\times \tan 2°\times ...\tan 89°$
$\Rightarrow \tan 1°\times \tan 89°\times \tan 2°\times \tan 88°\times ...\times \tan 44°\times \tan 46°\times \tan 45°$

We know that $\tan (90°-\theta )=\cot \theta .$

Therefore,
$\tan 1°=\tan (90°-89°)=\cot 89°$
$\tan 2°=\tan (90°-88°)=\cot 88°$
$\cdots$
Similarly, $\tan 44°=\cot 46°$

Now, we can write the given expression as,
$\cot 89°\times \tan 89°\times \cot 88°\times \tan 88°\times ...\times \cot 46°\times \tan 46°\times \tan 45°$

Also $\tan \theta \times \cot \theta =1$

So we get,
$\tan 1°\times \tan 2°\times ...\tan 89°=1$