First slide

Trigonometric equations

Question

The equation $2 \cos^{2} (\frac{x}{2}) \sin^{2} x = x^{2} + \frac{1}{x^{2}}, 0 \leq x \leq \frac{π}{2}$ has

Easy

A

one real solution
B

no solution
C

more than one real solution
D

two solutions

Solution

Since $x^{2} + x^{- 2} \geq 2$ and $2 \cos^{2} \frac{x}{2} \sin^{2} x \leq 2$
$\Rightarrow 2 \cos^{2} \frac{x}{2} \sin^{2} x = 2 \Rightarrow \cos^{2} \frac{x}{2} \sin^{2} x = 1 \Rightarrow \cos \frac{x}{2} = \sin x = 1 w h i c h i s n o t p o s s i b l e f o r a n y c h o i c e o f x .$

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