# The value of tan 1° tan 2° tan 3° ...... tan 89° is

Question:

The value of $\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ}$ $\tan 89^{\circ}$ is

(a) 1

(b) $-1$

(c) 0

(d) None of these

Solution:

Here we have to find: $\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ}$ $\tan 89^{\circ}$

$\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ}$ $\tan 89^{\circ}$

$=\tan \left(90^{\circ}-89^{\circ}\right) \tan \left(90^{\circ}-88^{\circ}\right) \tan \left(90^{\circ}-87^{\circ}\right) \ldots \tan 87^{\circ} \tan 88^{\circ} \tan 89^{\circ}$

$=\cot 89^{\circ} \cot 88^{\circ} \cot 87^{\circ} \ldots \tan 87^{\circ} \tan 88^{\circ} \tan 89^{\circ}$

$=\left(\cot 89^{\circ} \tan 89^{\circ}\right)\left(\cot 88^{\circ} \tan 88^{\circ}\right)\left(\cot 87^{\circ} \tan 87^{\circ}\right) \ldots\left(\cot 44^{\circ} \tan 44^{\circ}\right) \tan 45^{\circ}$

$\begin{array}{l}=1 \times 1 \times 1 \ldots 1 \times 1 \\ =1\end{array}$ [since $\left.\cot \theta \tan \theta=1\right]$