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Mathematics Question on Trigonometric Functions

The value of tan⁡81∘−tan⁡63∘−tan⁡27∘+tan⁡9∘\tan 81^{\circ} -\tan 63^{\circ} -\tan 27^{\circ} + \tan 9^{\circ} is equal to

A

1

B

2

C

3

D

4

Answer

4

Explanation

Solution

tan⁡81∘−tan⁡63∘−tan⁡27∘+tan⁡9∘\tan 81^{\circ} -\tan 63^{\circ} -\tan 27^{\circ} + \tan 9^{\circ} = \left\\{\tan \left(90^{\circ} -9^{\circ}\right) + \tan 9^{\circ}\right\\} - \left\\{\tan \left(90^{\circ} - 27^{\circ}\right) + \tan27^{\circ}\right\\} =(cot⁡9∘+tan⁡9∘)−(cot⁡27∘+tan⁡27∘) = \left(\cot 9^{\circ} + \tan 9^{\circ}\right) - \left(\cot 27 ^{\circ} + \tan27^{\circ}\right) =2cosec18∘−2cosec54∘= 2 cosec 18^{\circ} - 2 cosec 54^{\circ} (∵tan⁡θ+cot⁡θ=2cosec2θ)\left(\because \tan \theta + \cot \theta = 2 cosec 2\theta\right) =2sin⁡18∘−2sin⁡54∘=2sin⁡18∘−2cos⁡36∘= \frac{2}{ \sin 18^{\circ}} - \frac{2}{\sin54^{\circ}} = \frac{2}{\sin18^{\circ} } - \frac{2}{\cos 36^{\circ}} =2×45−1−2×45+1 = \frac{2\times4}{ \sqrt{5} - 1} - \frac{2\times4}{\sqrt{5} + 1 }