First slide

Multiple and sub- multiple Angles

Question

The value of $\tan 6^{\circ} \tan 42^{\circ} \tan 66^{\circ} \tan 78^{\circ}$ , is

Moderate

A

1
B

$1 / 2$
C

$1 / 4$
D

$1 / 8$

Solution

We have,
$\begin{array}{l} \tan 6^{\circ} \tan 42^{\circ} \tan 66^{\circ} \tan 78^{\circ} \\ = \tan 6^{\circ} \tan (60^{\circ} - 18^{\circ}) \tan (60^{\circ} + 6^{\circ}) \tan (60^{\circ} + 18^{\circ}) \\ = \frac{\tan 6^{\circ} \tan (60^{\circ} + 6^{\circ}) \tan 18^{\circ} \tan (60^{\circ} - 18^{\circ}) \tan (60^{\circ} + 18^{\circ})}{\tan 18^{\circ}} \\ = \frac{\tan 6^{\circ} \tan (60^{\circ} + 6^{\circ}) \tan (3 \times 18^{\circ})}{\tan 18^{\circ}} \\ = \frac{\tan 6^{\circ} \tan (60^{\circ} - 6) \tan (60^{\circ} + 6)}{\tan 18^{\circ}} = \frac{\tan 18^{\circ}}{\tan 18^{\circ}} = 1 \end{array}$

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