Question

Consider the following statements
Statement I $\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ} +\tan 81^{\circ}=4$
Statement II $\tan 9^{\circ}+\cot 9^{\circ}=\frac{2}{\sin 18^{\circ}}$ and $\cot 27^{\circ}+\tan 27^{\circ}=\frac{2^{\sin }}{\cos 36^{\circ}}$
Choose the correct option.

• A. Statement I is true; Statement II is true; Statement II is a correct explanation for Statement I
• B. Statement I is true; Statement II is true; Statement II is not a correct explanation for Statement I
• C. Statement I is true; Statement II is false
• D. Statement I is false; Statement II is true

Answer 1

Answer: (A)

We have $ \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}+\tan \left(90^{\circ}-9^{\circ}\right)-\tan 27^{\circ}-\tan \left(90^{\circ}-27^{\circ}\right) $
$ -\tan 9^{\circ}+\cot 9^{\circ}-\left(\tan 27^{\circ}+\cot 27^{\circ}\right) ....$(i)
Also,$ \tan 9^{\circ}+\cot 9^{\circ}=\frac{1}{\sin 9^{\circ} \cos 9^{\circ}}=\frac{2}{\sin 18^{\circ}} .....$(ii)
$\left(\because \frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}=\frac{1}{\sin x \cdot \cos x}\right)$
Similarly, $ \tan 27^{\circ}+\cot 27^{\circ}$
$=\frac{1}{\sin 27^{\circ} \cos 27^{\circ}}=\frac{2}{\sin 54^{\circ}}=\frac{2}{\cos 36^{\circ}} .....$(iii)
Using Eqs. (ii) and (iii) in Eq. (i), we get
$\tan 9^{\circ}- \tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ} $
$=\frac{2}{\sin 18^{\circ}}-\frac{2}{\cos 36^{\circ}} $
$ =\frac{2 \times 4}{\sqrt{5}-1}-\frac{2 \times 4}{\sqrt{5}+1}=4$