First slide

Multiple and sub- multiple Angles

Question

$\tan 9 ° - \tan 27 ° - \tan 63 ° + \tan 81 °$ is equal to

Moderate

A

0
B

1
C

-1
D

4

Solution

We have,
$\begin{array}{l} \tan 9^{\circ} - \tan 27^{\circ} - \tan 63^{\circ} + \tan 81^{\circ} \\ = (\tan 9^{\circ} + \tan 81^{\circ}) - (\tan 27^{\circ} + \tan 63^{\circ}) \\ = (\tan 9^{\circ} + \cot 9^{\circ}) - (\tan 27^{\circ} + \cot 27^{\circ}) \\ = \frac{1}{\sin 9^{\circ} \cos 9^{\circ}} - \frac{1}{\sin 27^{\circ} \cos 27^{\circ}} \\ = \frac{2}{\sin 18^{\circ}} - \frac{2}{\sin 54^{\circ}} \\ = 2 \frac{\sin 54^{\circ} - \sin 18^{\circ}}{\sin 54^{\circ} \sin 18^{\circ}} = 2 \frac{\cos 36^{\circ} - \sin 18^{\circ}}{\sin 18^{\circ} \cos 36^{\circ}} = 4 \end{array}$

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