Question

The value of $2\tan \frac{\pi }{10}+3\sec \frac{\pi }{10}-4\cos \frac{\pi }{10}$

• A. $0$
• B. $\sqrt{5}$
• C. $1$
• D. none of these

Answer 1

The correct option is A $0$

$a=2\tan \frac{\pi }{10}+3\sec \frac{\pi }{10}-4\cos \frac{\pi }{10}$
$\Rightarrow a=\frac{2\sin \frac{\pi }{10}+3-4{\cos }^2\frac{\pi }{10}}{\cos \frac{\pi }{10}}$
$\Rightarrow a=\frac{2\sin \frac{\pi }{10}\cos \frac{\pi }{10}-(4{\cos }^3\frac{\pi }{10}-3\cos \frac{\pi }{10})}{{\cos }^2\frac{\pi }{10}}$
As we know that $2\sin \theta \cos \theta =\sin 2\theta \&\cos 3\theta =4{\cos }^3\theta -3\cos \theta$
$\Rightarrow a=\frac{\sin \frac{\pi }{5}-\cos \frac{3\pi }{10}}{{\cos }^2\frac{\pi }{10}}$
$\Rightarrow a=\frac{\sin \frac{\pi }{5}-\sin (\frac{\pi }{2}-\frac{3\pi }{10})}{{\cos }^2\frac{\pi }{10}}=\frac{\sin \frac{\pi }{5}-\sin \frac{\pi }{5}}{{\cos }^2\frac{\pi }{10}}$
$\therefore a=0$
Ans: A