Evaluate:
$ \tan 7^{\circ} \tan 23^{\circ} \tan 60^{\circ} \tan 67^{\circ} \tan 83^{\circ} $

Given:

$\tan 7^{\circ} \tan 23^{\circ} \tan 60^{\circ} \tan 67^{\circ} \tan 83^{\circ}$.

To do:

We have to evaluate $\tan 7^{\circ} \tan 23^{\circ} \tan 60^{\circ} \tan 67^{\circ} \tan 83^{\circ}$.

Solution:  

We know that,

$tan\ (90^{\circ}- \theta) = cot\ \theta$

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

Therefore,

$\tan 7^{\circ} \tan 23^{\circ} \tan 60^{\circ} \tan 67^{\circ} \tan 83^{\circ}=\tan 7^{\circ} \tan 23^{\circ} \tan 60^{\circ} \tan (90^{\circ}- 23^{\circ}) \tan (90^{\circ}-7^{\circ})$

$=\tan 7^{\circ} \tan 23^{\circ} (\sqrt3) \cot 23^{\circ} \cot 7^{\circ}$     (Since $\tan 60^{\circ}=\sqrt3$)

$=\sqrt3(\tan 7^{\circ} \cot 7^{\circ})(\tan 23^{\circ}\cot 23^{\circ})$

$=\sqrt3\times1\times1$

$=\sqrt3$ 

Hence, $\tan 7^{\circ} \tan 23^{\circ} \tan 60^{\circ} \tan 67^{\circ} \tan 83^{\circ}=\sqrt3$. 

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Updated on: 10-Oct-2022

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