The Value of \(2\tan\frac{\pi}{10}+3\sec\frac{\pi}{10}-4\cos\frac{\pi}{10}\)
Question
Find the value of
\[ 2\tan\frac{\pi}{10} + 3\sec\frac{\pi}{10} – 4\cos\frac{\pi}{10} \]
(a) \(0\)
(b) \(\sqrt5\)
(c) \(1\)
(d) none of these
Solution
Let \[ \theta=\frac{\pi}{10} \]
Then \[ 2\tan\theta+3\sec\theta-4\cos\theta = \frac{2\sin\theta}{\cos\theta} +\frac{3}{\cos\theta} -4\cos\theta \]
Taking LCM,
\[ = \frac{2\sin\theta+3-4\cos^2\theta}{\cos\theta} \]
Using \[ 4\cos^2\theta=2(1+\cos2\theta) \]
\[ = \frac{2\sin\theta+1-2\cos2\theta}{\cos\theta} \]
Since \[ \theta=\frac{\pi}{10} \Rightarrow 2\theta=\frac{\pi}{5} \] and \[ 2\cos\frac{\pi}{5} = \frac{\sqrt5+1}{2} \]
Also, \[ \sin\frac{\pi}{10} = \frac{\sqrt5-1}{4} \]
Substituting,
\[ 2\sin\frac{\pi}{10}+1-2\cos\frac{\pi}{5} = \frac{\sqrt5-1}{2}+1-\frac{\sqrt5+1}{2} =0 \]
Therefore,
\[ 2\tan\frac{\pi}{10} + 3\sec\frac{\pi}{10} – 4\cos\frac{\pi}{10} =0 \]
Final Answer
\[ \boxed{0} \]
Hence, the correct option is (a) 0.