Question

The smallest positive number p for which the equation $\cos \left(p\sin x\right)=\sin \left(p\cos x\right)$ has a solution $x\in [0,2\pi ]$

• A. 2
• B. $\pi \sqrt{2}/4$
• C. $\pi /2$
• D. none of these

Answer 1

The correct option is B $\pi \sqrt{2}/4$

we have, $\cos \left(p\sin x\right)=\sin \left(p\cos x\right)$

We have $\cos \left(p\sin x\right)=\cos (\frac{\pi }{2}-p\cos x)$

$\Rightarrow \frac{\pi }{2}-p\cos x=2n\pi \pm p\sin x,n\in I$

$\Rightarrow p(\cos x\pm \sin x)=\frac{\pi }{2}-2n\pi$

$\Rightarrow \cos x\pm \sin x=(1-4n)\frac{\pi }{2p}$

Since $-\sqrt{2}\le \cos x\pm \sin x\le \sqrt{2}$

$-\sqrt{2}\le (1-4n)\frac{\pi }{2p}\le \sqrt{2}$

$|(1-4n)\frac{\pi }{2p}|\le \sqrt{2}$

$\Rightarrow |p|\ge |(1-4n)|\frac{\pi }{2\sqrt{2}}$

The smallest positive value of p is $\frac{\pi }{2\sqrt{2}}=\frac{\pi \sqrt{2}}{4}$