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  4. The value of tan 10° tan 20° tan 30° tan 40° tan 50° tan 60° tan 70° tan 80° =

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Q. The value of $\tan 10^{\circ}\, \tan 20^{\circ} \, \tan 30^{\circ} \, \tan 40^{\circ} \, \tan 50^{\circ}\, \tan 60^{\circ} \tan 70^{\circ} \, \tan 80^{\circ} =$

COMEDKCOMEDK 2012Trigonometric Functions

Solution:

$\tan 10^{\circ}\, \tan 20^{\circ} \, \tan 30^{\circ} \, \tan 40^{\circ} \, \tan 50^{\circ}\, \tan 60^{\circ} \tan 70^{\circ} \, \tan 80^{\circ}$
$ = \tan (90 - 80)^{\circ} \tan (90 - 70)^{\circ} \tan (90 - 60)^{\circ} \tan (90 - 50)^{\circ} \tan 50^{\circ} \tan 60^{\circ} \tan 70^{\circ} \tan 80^{\circ}$
$= \tan 80^{\circ} \times \cot 80^{\circ} \times \tan 70^{\circ} \times \cot 70^{\circ} \times \tan 60^{\circ} \times \cot 60^{\circ} \times \tan 50^{\circ} \times \cot 50^{\circ} = 1 $