Question : What is the value of $\left(\tan 10^{\circ} \tan 80^{\circ}+\tan 20^{\circ} \tan 70^{\circ}+\tan 30^{\circ} \tan 60^{\circ}+\tan 40^{\circ} \tan 50^{\circ}\right)$?
Option 1: 4
Option 2: 3
Option 3: 5
Option 4: 2
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Correct Answer: 4
Solution :
Given: $(\tan 10^{\circ} \tan 80^{\circ}+\tan 20^{\circ} \tan 70^{\circ}+\tan 30^{\circ} \tan 60^{\circ}+\tan 40^{\circ} \tan 50^{\circ})$
$=\tan 10^{\circ} \tan (90^{\circ}-10^{\circ})+\tan 20^{\circ} \tan (90^{\circ}-20^{\circ})+\tan 30^{\circ} \tan (90^{\circ}-30^{\circ})+\tan 40^{\circ} \tan (90^{\circ}-40^{\circ})$
$=\tan 10^{\circ} \cot 10^{\circ}+\tan 20^{\circ} \cot 20^{\circ}+\tan 30^{\circ} \cot 30^{\circ}+\tan 40^{\circ} \cot 40^{\circ}$
$=1+1+1+1$
$=4$
Hence, the correct answer is 4.
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