Question

The value of $2\tan \frac{\pi }{10}+3\sec \frac{\pi }{10}-4\cos \frac{\pi }{10}$ is
(a) 0
(b) $\sqrt{5}$
(c) 1
(d) none of these

Answer 1

(a) 0

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ 2\tan \frac{\mathrm{\pi }}{10}+3\sec \frac{\mathrm{\pi }}{10}-4\cos \frac{\mathrm{\pi }}{10} \\ =2\tan 18°+3\sec 18°-4\cos 18° \\ =2\frac{\sin 18°}{\cos 18°}+3\times \frac{1}{\cos 18°}-4\cos 18° \\ =2\times \frac{{\displaystyle \frac{\sqrt{5}-1}{4}}}{{\displaystyle \frac{\sqrt{10+2\sqrt{5}}}{4}}}+3\times \frac{1}{{\displaystyle \frac{\sqrt{10+2\sqrt{5}}}{4}}}-4\times \frac{\sqrt{10+2\sqrt{5}}}{4} \\ =2\times \frac{\sqrt{5}-1}{\sqrt{10+2\sqrt{5}}}+3\times \frac{4}{\sqrt{10+2\sqrt{5}}}-\sqrt{10+2\sqrt{5}} \\ =\frac{2\sqrt{5}-2+12-{\left(\sqrt{10+2\sqrt{5}}\right)}^2}{\left(\sqrt{10+2\sqrt{5}}\right)} \\ =\frac{2\sqrt{5}+10-10-2\sqrt{5}}{\left(\sqrt{10+2\sqrt{5}}\right)} \\ =0 \end{gathered}$$