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Trigonometrical Equations
normal
સમીકરણ $2{\cos ^2}\left( {\frac{x}{2}} \right)\,{\sin ^2}x\, = \,{x^2}\, + \,\frac{1}{{{x^2}}},\,0\,\, \leqslant \,\,x\,\, \leqslant \,\,\frac{\pi }{2}\,\,$ ના ............... ઉકેલો મેળવો
A
શૂન્ય
B
એક વાસ્તવિક
C
એક કરતાં વધારે વાસ્તવિક
D
એક પણ નહીં
Solution
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 $\neq 2$
$\Rightarrow$ No solution.
Standard 11
Mathematics
