MCQ
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$1$
- DNone of these
Solution:
We have,
$2\tan\frac{\pi}{10}+3\sec\frac{\pi}{10}-4\cos\frac{\pi}{10}$
$=2\tan18^\circ+3\sec18^\circ-4\cos18^\circ$
$=2\frac{\sin18^\circ}{\cos18^\circ}+3\times\frac{1}{\cos18^\circ}-4\cos18^\circ$
$=2\times\frac{\frac{\sqrt{5}-1}{4}}{\frac{\sqrt{10+2\sqrt{5}}}{4}}+3\times\frac{1}{\frac{\sqrt{10+2\sqrt{5}}}{4}}=4\times\frac{\sqrt{10+2\sqrt{5}}}{4}$
$=2\times\frac{\sqrt{5}-1}{\sqrt{10+2\sqrt{5}}}+3\times\frac{4}{\sqrt{10+2\sqrt{5}}}-\sqrt{10+2\sqrt{5}}$
$=\frac{2\sqrt{5}-2+12-\Big(\sqrt{10+2\sqrt{5}}\Big)^2}{\Big(\sqrt{10+2\sqrt{5}}\Big)}$
$=\frac{2\sqrt{5}+10-10-2\sqrt{5}}{\Big(\sqrt{10+3\sqrt{5}}\Big)}$
$=0$
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$\text{x}\in[-10, 10)$