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
(a) 0
We have,
$2 \tan \frac{\pi}{10}+3 \sec \frac{\pi}{10}-4 \cos \frac{\pi}{10}$
$=2 \tan 18^{\circ}+3 \sec 18^{\circ}-4 \cos 18^{\circ}$
$=2 \times \frac{\frac{\sqrt{5}-1}{4}}{\frac{\sqrt{10+2 \sqrt{5}}}{4}}+3 \times \frac{1}{\frac{\sqrt{10+2 \sqrt{5}}}{4}}-4 \times \frac{\sqrt{10+2 \sqrt{5}}}{4}$
$=2 \times \frac{\sqrt{5}-1}{\sqrt{10+2 \sqrt{5}}}+3 \times \frac{4}{\sqrt{10+2 \sqrt{5}}}-\sqrt{10+2 \sqrt{5}}$
$=\frac{2 \sqrt{5}-2+12-(\sqrt{10+2 \sqrt{5}})^{2}}{(\sqrt{10+2 \sqrt{5}})}$
$=\frac{2 \sqrt{5}+10-10-2 \sqrt{5}}{(\sqrt{10+2 \sqrt{5}})}$
$=0$
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