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Question
The value of \[2 \tan \frac{\pi}{10} + 3 \sec \frac{\pi}{10} - 4 \cos \frac{\pi}{10}\] is
Options
0
- \[\sqrt{5}\]
1
none of these
Solution
\[\text{ We have } , \]
\[2\tan\frac{\pi}{10} + 3\sec\frac{\pi}{10} - 4\cos\frac{\pi}{10}\]
\[ = 2\tan18° + 3\sec18° - 4\cos18° \]
\[ = 2\frac{\sin18° }{\cos18° } + 3 \times \frac{1}{\cos18° } - 4\cos18° \]
\[ = 2 \times \frac{\frac{\sqrt{5} - 1}{4}}{\frac{\sqrt{10 + 2\sqrt{5}}}{4}} + 3 \times \frac{1}{\frac{\sqrt{10 + 2\sqrt{5}}}{4}} - 4 \times \frac{\sqrt{10 + 2\sqrt{5}}}{4}\]
\[ = 2 \times \frac{\sqrt{5} - 1}{\sqrt{10 + 2\sqrt{5}}} + 3 \times \frac{4}{\sqrt{10 + 2\sqrt{5}}} - \sqrt{10 + 2\sqrt{5}}\]
\[ = \frac{2\sqrt{5} - 2 + 12 - \left( \sqrt{10 + 2\sqrt{5}} \right)^2}{\left( \sqrt{10 + 2\sqrt{5}} \right)}\]
\[ = \frac{2\sqrt{5} + 10 - 10 - 2\sqrt{5}}{\left( \sqrt{10 + 2\sqrt{5}} \right)}\]
\[ = 0\]
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