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Trigonometrical Equations
hard
यदि $X=\{x \in R : \cos (\sin x)=\sin (\cos x)\}$, तो $X$ में कुल अवयवों की संख्या
A
$0$
B
$2$
C
$4$
D
परिमित नहीं है.
(KVPY-2016)
Solution
(a)
We have,
$X=\{x \in R: \cos (\sin x)=\sin (\cos x)\}$
$\cos (\sin x)=\sin (\cos x)$
$\quad \sin \left(\frac{\pi}{2} \pm \sin x\right)=\sin (\cos x)$
$\Rightarrow \quad \cos x=\frac{\pi}{2} \pm \sin x$
$\Rightarrow \cos x=n \pi+(-1)^n\left(\frac{\pi}{2}+\sin x\right), n \in I$
$\Rightarrow \cos x \pm \sin x=n \pi+(-1)^n \frac{\pi}{2}, n \in I$
As $L H S \in[-\sqrt{2}, \sqrt{2}]$ and it does not satisfy the RHS.
$\therefore$ No solution exists.
Standard 11
Mathematics
