The equation 2{cos ^2}left( {frac{x}{2}} righ | vedclass

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Question

Equation is: $2 \cos ^{2} \frac{x}{2} \sin ^{2} x=x^{2}+\frac{1}{x^{2}}$

$R . H.S =x^{2}+\frac{1}{x^{2}}$

$x^{2}+\frac{1}{x^{2}} \geq 2 \forall

Vedclass · Accepted Answer

no solution

Vedclass · Answer

Equation is: $2 \cos ^{2} \frac{x}{2} \sin ^{2} x=x^{2}+\frac{1}{x^{2}}$

$R . H.S =x^{2}+\frac{1}{x^{2}}$

$x^{2}+\frac{1}{x^{2}} \geq 2 \forall x \in R-\{0\}$

$[$ using $A M \geq G M]$

Also $x^{2}+\frac{1}{x^{2}}=2$ at $x=\pm 1$

$LH.S =2 \cos ^{2} \frac{x}{2} \sin ^{2} x$

 

Max value of $\cos$ or $\sin$ is 1

$\Rightarrow 2 \cos ^{2} \frac{x}{2} \sin ^{2} x \leq 2$

$\Rightarrow$ L.H.S $=$ R.H.S only if both are $=2$

$\Rightarrow x=1$

but at $x=1,$ L.H.S $
eq 2$

$\Rightarrow$ No solution.

The equation $2{\cos ^2}\left( {\frac{x}{2}} \right)\,{\sin ^2}x\, = \,{x^2}\, + \,\frac{1}{{{x^2}}},\,0\,\, \leqslant \,\,x\,\, \leqslant \,\,\frac{\pi }{2}\,\,$ has

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