Q66 / 160 · TS EAMCET 2004 · Mathematics

The expression $\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}$ is equal to

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

Solution

$\begin{aligned} & \tan 9^{\circ}-\tan 27^{\circ}-\cot 27^{\circ}+\cot 9^{\circ} \\ & =\left(\frac{\sin 9^{\circ}}{\cos 9^{\circ}}+\frac{\cos 9^{\circ}}{\sin 9^{\circ}}\right)-\left(\frac{\sin 27^{\circ}}{\cos 27^{\circ}}+\frac{\cos 27^{\circ}}{\sin 27^{\circ}}\right) \\ & =\left(\frac{\sin ^2 9^{\circ}+\cos ^2 9^{\circ}}{\cos 9^{\circ}}\right)-\left(\frac{\sin ^2 27^{\circ}+\cos ^2 27^{\circ}}{\cos 27^{\circ} \sin 27^{\circ}}\right) \\ & =\frac{2}{\sin 18^{\circ}}-\frac{2}{\sin 54^{\circ}} \\ & =\frac{2}{\frac{\sqrt{5}-1}{4}}-\frac{2}{\frac{\sqrt{5}+1}{4}}=\frac{8}{\sqrt{5}-1}-\frac{8}{\sqrt{5}+1} \\ & =\frac{16}{5-1}=\frac{16}{4}=4\end{aligned}$

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

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