Number of real solutions x of equation cos ^2(x sin (2 x))+frac{1}{1+x^2}=cos ^2 x+sec ^2 x is

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Trigonometrical Equations

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The number of real solutions $x$ of the equation $\cos ^2(x \sin (2 x))+\frac{1}{1+x^2}=\cos ^2 x+\sec ^2 x$ is

$0$

$1$

$2$

infinite

(KVPY-2018)

(b)

We have,

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

$\text { LHS }=\cos ^2(x \sin 2 x)+\frac{1}{1+x^2}$

$\cos ^2(x \sin 2 x)+\frac{1}{1+x^2} \leq 2$

$\text { RHS }=\cos ^2 x+\sec ^2 x \geq 2$

$\qquad \quad \operatorname{LHS}^2 x+\sec ^2 x=2 \text { at } x=0$

Hence, only one solution.

Standard 11

Mathematics

medium

(IIT-2004)

hard

(JEE MAIN-2023)

medium

easy

normal

(IIT-2015)