Q152 / 270 · NDA 2011 (Phase 2) · Mathematics

What is the value of $\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ} ?$

  • A
    $1$
  • B
    $2$
  • C
    $3$
  • Correct answer
    D
    $4$

Solution

$\begin{aligned} & \text { We have, } \tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ} \\ & =\left(\tan 9^{\circ}+\tan 81^{\circ}\right)-\left(\tan 27^{\circ}+\tan 63^{\circ}\right) \\ & =\frac{1}{\cos 9^{\circ} \cdot \cos 81^{\circ}}-\frac{1}{\cos 27^{\circ} \cdot \cos 63^{\circ}} \\ & =\frac{1}{\sin 9^{\circ} \cdot \cos 9^{\circ}}-\frac{1}{\sin 27^{\circ} \cdot \cos 27^{\circ}} \\ & =\frac{2}{\sin 18^{\circ}}-\frac{2}{\sin 54^{\circ}} \\ & =2\left[\frac{\sin 54^{\circ}-\sin 18^{\circ}}{\sin 54^{\circ} \cdot \sin 18^{\circ}}\right] \\ & =2\left[\frac{2 \cos 36^{\circ} \cdot \sin 18^{\circ}}{\sin 18^{\circ} \cdot \cos 36^{\circ}}\right]=2 \times 2=4 .\end{aligned}$

Hence, the correct answer is option: 'D'.

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