Relating Trigonometry And Rational Functions

Geometry Level 2

$2 \cos^{2}\frac{x}{2} \sin^{2}x = x^{2} + \frac{1}{x^{2}}$

where $0 \leq x \leq \frac{\pi}{2}$ . What is the number of real root(s) of the above equation?

0 2 4 6

1 solution

Aditya Chauhan
Jul 6, 2016

Relevant wiki: Applying the Arithmetic Mean Geometric Mean Inequality

Using A.M. $\geq$ G.M. , the minimum value of RHS is 2 which occurs only at $x=1$ but LHS at this point is not equal to RHS .Also LHS can never be greater than 2. Therefore the given equation has no solution.

Max value of LHS is coming out to be 32/27. Could you please give the detailed solution ? Thanks.

Tushar Jawalia - 4 years, 11 months ago

How did you get 32/27?

Atomsky Jahid - 4 years, 11 months ago

Max value of LHS is 1. Good solution btw

Harshit Singhania - 4 years, 11 months ago

It is not 1. It is greater than 1.

Aditya Chauhan - 4 years, 11 months ago

If u add and subtract (sin x) ^2, then u end up with 1 - (cos x) ^4

Harshit Singhania - 4 years, 11 months ago

@Harshit Singhania – No sorry my bad.

Harshit Singhania - 4 years, 11 months ago

@Harshit Singhania – No problem.

Aditya Chauhan - 4 years, 11 months ago

Yeah indeed the shortest way , we can also do it by checking quadratic discriminant ...

A Former Brilliant Member - 4 years, 11 months ago

Relating Trigonometry And Rational Functions

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