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Maths-

General

Easy

Question

The equation $2 cos to the power of 2 end exponent invisible function application fraction numerator x over denominator 2 end fraction sin to the power of 2 end exponent invisible function application x equals x to the power of 2 end exponent equals x to the power of negative 2 end exponent semicolon 0 less than x less or equal than fraction numerator pi over denominator 2 end fraction$ has

No real solution
One real solution
More than one solution
None of these

The correct answer is: No real solution

The given equation is
$2 cos to the power of 2 end exponent invisible function application open parentheses fraction numerator x over denominator 2 end fraction close parentheses sin to the power of 2 end exponent invisible function application x equals x to the power of 2 end exponent plus fraction numerator 1 over denominator x to the power of 2 end exponent end fraction$
Where $0 less than x less or equal than fraction numerator pi over denominator 2 end fraction$
$L. H. S. equals 2 cos to the power of 2 end exponent invisible function application open parentheses fraction numerator x over denominator 2 end fraction close parentheses sin to the power of 2 end exponent invisible function application x equals open parentheses 1 plus cos invisible function application x close parentheses sin to the power of 2 end exponent invisible function application x$
$because 1 plus cos invisible function application x less than 2$ and $sin to the power of 2 end exponent invisible function application x less or equal than 1$ for $0 less than x less than fraction numerator pi over denominator 2 end fraction$
$therefore open parentheses 1 plus cos invisible function application x close parentheses sin to the power of 2 end exponent invisible function application x less than 2$
and R.H.S. $equals x to the power of 2 end exponent plus fraction numerator 1 over denominator x to the power of 2 end exponent end fraction greater or equal than 2$
$therefore$ For $0 less than x less or equal than fraction numerator pi over denominator 2 end fraction$ , given equation is not possible for any real value of $x$

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