Question

The value of x which satisfy $\sqrt{17se{c}^{2}x+16(\frac{1}{2}tanxsecx-1)}=2tan(1+4sinx)$ is

• A.
• B.
• C.
• D. Infinite

Answer 1

The correct option is A

The given equation is
$\sqrt{17se{c}^{2}x+16(\frac{1}{2}tanxsecx-1)}$
=$2tanx(1+4sinx)$.....(1)
or, $\sqrt{16ta{n}^{2}x+8tanxsecx+se{c}^{2}x}$
=2tanx(1+4 sinx)
or, 4 tanx + secx = 2tanx(1+4sinx)
or, 8 $si{n}^{2}x-2sinx-1=0$
$\therefore sinx=\frac{1}{2}$ and $sinx=-\frac{1}{4}$
or, sinx=sin$\frac{\pi }{6}$ and $x=si{n}^{-1}(-\frac{1}{4})$
$\therefore$ Solution of (1) are
$x=n\pi +(-1{)}^n\frac{\pi }{6}$ and
$x=n\pi +(-1{)}^{n+1}si{n}^{-1}(\frac{1}{4})$ where $n\in I$.