Question

The value of $\tan 1°\tan 2°\tan 3°...\tan 89°$ is

(a) 0 (b) 1 (c) $\frac{1}{2}$ (d) not defined

Answer 1

We know that, $\tan \left(90°-\theta \right)=\cot \theta$

So,

$$\begin{gathered} \tan 89°=\tan \left(90°-1°\right)=\cot 1° \\ \tan 88°=\tan \left(90°-2°\right)=\cot 2° \\ \tan 87°=\tan \left(90°-3°\right)=\cot 3° \\ .... \\ .... \\ \tan 46°=\tan \left(90°-44°\right)=\cot 44° \end{gathered}$$

$$\begin{gathered} \therefore \tan 1°\tan 2°\tan 3°...\tan 89° \\ =\tan 1°\tan 2°\tan 3°...\tan 44°\tan 45°\tan 46°...\tan 87°\tan 88°\tan 89° \\ =\tan 1°\tan 2°\tan 3°...\tan 44°\tan 45°\cot 44°...\cot 3°\cot 2°\cot 1° \\ =\left(\tan 1°\cot 1°\right)\left(\tan 2°\cot 2°\right)\left(\tan 3°\cot 3°\right)...\left(\tan 44°\cot 44°\right)\tan 45° \\ =1\left(\tan 45°=1\mathrm{and}\tan \theta \cot \theta =1\right) \end{gathered}$$

Hence, the correct answer is option B.