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
$\tan 9^{\circ }-\tan 27^{\circ }-\tan 63^{\circ }+\tan 81^{\circ }$
$=\frac{2}{\sin 18^{\circ }}-\frac{2}{\cos 36^{\circ }}$
$=\frac{2\times 4}{\sqrt{5}-1}-\frac{2\times 4}{\sqrt{5}+1}=4$