Question

If $|2\sin \theta -\csc \theta |\ge 1$ and $\theta \ne \frac{n\pi }{2},n\in z$, then

• A. $\cos 2\theta \ge \frac{1}{2}$
• B. $\cos 2\theta \ge \frac{1}{4}$
• C. $\cos 2\theta \le \frac{1}{2}$
• D. $\cos 2\theta \le \frac{1}{4}$

Answer 1

Answer: (A)

$\cos 2\theta \ge \frac{1}{2}$

$\mid 2\sin \theta -\csc \theta \mid \ge 1$

Case 1:-

$2\sin \theta -\csc \theta \ge 1$

or $2{\sin }^2\theta -\sin \theta -1\ge 0$

or $2{\sin }^2\theta -2\sin \theta +\sin \theta -1\ge 0$

or, $2\sin \theta (\sin \theta -1)+1(\sin \theta -1)\ge 0$

or, $(\sin \theta -1)(2\sin \theta +1)\ge 0$

or, $\sin \theta \ge 1,\sin \theta \ge \frac{-1}{2}$

and $\sin \theta \le 1,\sin \theta \le \frac{-1}{2}$

Case 2:-

$2\sin \theta -\csc \theta \le -1$

$2{\sin }^2\theta -1+\sin \theta \le 0$

or, $2{\sin }^2\theta +2\sin \theta -\sin \theta -1\le 0$

or, $(2\sin \theta -1)(\sin \theta +1)\le 0$

or, $\sin \theta \le \frac{1}{2}$ and $\sin \theta \ge -1$

or, $\sin \theta \ge \frac{1}{2}$ and $\sin \theta \le -1$

Both case is possible when,

$\frac{-1}{2}\le \sin \theta \le \frac{1}{2}$

or, ${\sin }^2\theta \le \frac{1}{4}$

or, $2{\sin }^2\theta \le \frac{1}{2}$

or, $-2{\sin }^2\theta \ge \frac{-1}{2}$

or, $1-2{\sin }^2\theta \ge 1-\frac{1}{2}$

or, $\cos 2\theta \ge \frac{1}{2}$