First slide

Theory of expressions

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

Moderate

A

no real solution
B

one real solution
C

more than one real solution
D

none of these

Solution

Clearly ,
$LHS = 2 \cos^{2} (x / 2) \sin^{2} x \leq 2$ and, $RHS = x^{2} + \frac{1}{x^{2}} \geq 2$
Thus, the equality holds when each side is equal to 2.
But, RHS is equal to 2 for $x = 1$ while LHS is less than 2 for this value of $x$ .
Consequently the equation has no solution.

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