Question

The equation $2{\cos }^2\frac{x}{2}{\sin }^{2}x=x^2+\frac{1}{x^2},0\le x\le \frac{\pi }{2}$ has

• A. One real solution
• B. No real solution
• C. More than one real solution
• D. two real solutions

Answer 1

The correct option is B No real solution

$2{\cos }^2\frac{x}{2}{\sin }^{2}x=x^2+\frac{1}{x^2}$
$(1+cosx){\sin }^{2}x=x^2+\frac{1}{x^2}$
$\because (2{\cos }^2\frac{\Theta }{2}=1+\cos \Theta )$
${\sin }^{2}x+{\sin }^{2}x+\cos x=x^2+\frac{1}{x^2}$
max value of L.H.S is 1.
min value of $x^2+\frac{1}{x^2}is2asAM⩾GM$
$\therefore$ there is no point at which both become equal.