Needed a short method

Prove that $sin(cosx)=cos(sinx)$ has no real solution.

The method i came across was this

LHS is not equal to RHS.

If u have got any other method do post it.

#Geometry

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

Look at the function $f(x) = \cos(\sin(x)) - \sin(\cos(x)).$ This function is even and periodic with period $2\pi.$ so we just need to focus on the interval $[0, \pi].$

Now $f(0) = 1 \gt 0.$ Also, for $x \in [\frac{\pi}{2}, \pi]$ we have $\cos(\sin(x)) \gt 0$ and $\sin(\cos(x)) \le 0,$ so $f(x) \gt 0$ on this interval.

So we now just need to focus on the interval $(0, \frac{\pi}{2}).$ On this interval we have that both $\cos(x)$ and $\sin(x)$ lie between $0$ and $1,$ for which $\sin(\cos(x)) \lt \cos(x) \lt \cos(\sin(x)),$ , i.e., $f(x) \gt 0.$ Thus $f(x) \gt 0$ for all $x$ .

(To expand on this last step, note that for $\theta$ in $(0, \frac{\pi}{2})$ we have $0 \lt \sin(\theta) \lt \theta,$ and so $\cos(\sin(\theta)) \gt \cos(\theta).$ )

Brian Charlesworth - 6 years, 1 month ago

Yes! I recall when I first saw bounding those values by $\cos x$ , which seemed really ingenious to me.

Calvin Lin Staff - 6 years, 1 month ago

The easiest way for me is to plot the graph for the full range of $x \in [0,2\pi)$ . The graph is as follows. We note that the $\sin{(\cos{x})} \ne \cos{(\sin{x})}$ always.

Chew-Seong Cheong - 6 years, 1 month ago

Well, part of the question would then be "How do you know that's how the graphs should be drawn (Other than relying on a calculator / graphing capabilities)?"

Calvin Lin Staff - 6 years, 1 month ago

Do you mean how to do graphs with Excel spreadsheet? If so, maybe I should write a wiki on it. There are others requested for it. I just don't like to do it.

Chew-Seong Cheong - 6 years, 1 month ago

It is easy to see that $cos(cos(x))-sin(sin(x))=0$ has no solutions. Just replace $x$ with $\frac{\pi} {2}-x$

Raghav Vaidyanathan - 6 years, 1 month ago

@Brian Charlesworth @Tanishq Varshney

Raghav Vaidyanathan - 6 years, 1 month ago

It's convincing that $cos(cos(x)) \ne sin(sin(x))$ but how can you prove that statement?

Curtis Clement - 6 years, 1 month ago

Sketch the rough graph of both the functions

Nikhil Tanwar - 6 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$