Let X - x - R - cos sin x -sin cos x - The number of elements in X isA- 0B- 2C- 4D- not finite

The correct option is A 0cos (sin x) = sin(cos x) ⇒sin(π/2±sin x)= sin(cos x) ⇒cos x=nπ± ( 1)^n (π/2±sin x), n ∈ I ⇒cos x ±sin x=nπ+( 1)^nπ/2 , n∈ I As L ...

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Standard XII

Mathematics

Trigonometric Ratios of Compound Angles

Let X = x ∈ R...

Question

Let $X = {x \in R : cos (sin x) = sin (cos x)}$ . The number of elements in $X$ is

0

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2

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4

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not finite

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Solution

The correct option is A $0$
$cos (sin x) = sin (cos x) \Rightarrow sin (\frac{π}{2} \pm sin x) = sin (cos x) \Rightarrow cos x = n π \pm (- 1)^{n} (\frac{π}{2} \pm sin x), n \in I \Rightarrow cos x \pm sin x = n π + (- 1)^{n} \frac{π}{2}, n \in I$

As LHS $\in [- \sqrt{2}, \sqrt{2}]$ , and it does not satisfy RHS.
Hence no solution possible.

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Let X={x∈R:cos(sinx)=sin(cosx)}. The number of elements in X is

The correct option is A 0cos(sinx)=sin(cosx)⇒sin(π2±sinx)=sin(cosx)⇒cosx=nπ±(−1)n(π2±sinx), n∈I⇒cosx±sinx=nπ+(−1)nπ2, n∈I As LHS ∈[−√2,√2], and it does not satisfy RHS. Hence no solution possible.

Let $X = {x \in R : cos (sin x) = sin (cos x)}$ . The number of elements in $X$ is