Question

The value's of $x\in (0,\frac{\pi }{2})$ satisfying $\frac{\sqrt{3}-1}{\sin x}+\frac{\sqrt{3}+1}{\cos x}=4\sqrt{2}$ are

• A. $\frac{\pi }{12}$
• B. $\frac{5\pi }{12}$
• C. $\frac{7\pi }{24}$
• D. $\frac{11\pi }{36}$

Answer 1

Answer: (A), (D)

$\frac{11\pi }{36}$

$\frac{\sqrt{3}-1}{2\sqrt{2}\sin x}+\frac{\sqrt{3}+1}{2\sqrt{2}\cos x}=2$
$\sin \frac{\pi }{12}\cos x+\cos \frac{\pi }{12}\sin x=\sin 2x$
$\sin 2x=\sin (x+\frac{\pi }{12})$
$\therefore 2x=x+\frac{\pi }{12}\text{or}2x=\pi -(x+\frac{\pi }{12})(\because 2x\in (0,\pi \left)\right)$
$x=\frac{\pi }{12}\text{or}3x=\frac{11\pi }{12}$
$\therefore x=\frac{\pi }{12}\text{or}\frac{11\pi }{36}$