Question

The minimum value of $\cos \theta +\sin \theta +\frac{2}{\sin 2\theta }$ for $\theta ϵ(0,\pi /2)$ is

• A. $2+\sqrt{2}$
• B. $2$
• C. $1+sqrt2$
• D. $2\sqrt{2}$

Answer 1

The correct option is A $2+\sqrt{2}$

$\sin \theta +\cos \theta +\frac{1}{\sin \theta .\cos \theta }$
As the expression remain unchanged by interchanging $\sin \theta$ and $\cos \theta$ so minimum is achieved for
$\sin \theta =\cos \theta$ so $\theta =\frac{\pi }{4}$ in $(0,\frac{\pi }{2})$
so, minimum value $=2+\sqrt{2}$