<s>[INST] Does the equation $2\cos^2 (x/2) \sin^2 (x/2) = x^2+\frac{1}{x^2}$ have real solution? 
Do the equation
  $$2\cos^2 (x/2) \sin^2 (x/2) = x^2+\frac{1}{x^2}$$
  have any real solutions?

Please help. This is an IITJEE question.
Here $x$ is an acute angle.
I cannot even start to attempt this question. I cannot understand.  
 [/INST] the right hand of the equation, you have $$x^2 + \frac1{x^2} = \left(x-\frac 1x\right)^2 + 2 \ge 2 \tag 1$$and equality occurs for $$x = \pm 1.$$
on the left hand side, we have $$ \frac12 \sin^2 x = 2\cos^2 (x/2) \sin^2 (x/2) \le \frac12 \tag 2$$ equality for $$\sin x = \pm 1.$$
but $(1)$ and $(2)$ are inconsistent, therefore $$2\cos^2 (x/2) \sin^2 (x/2) = x^2+\frac{1}{x^2} $$ has no real solutions.
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