Question

prove that equation cos(sin x)=sin(cos x) does not possess real roots

Answer 1

:)

Answer 2

Blah. I would consider the function

f(x) = cos(sin x) - sin(cos x)

In the first pass at this problem I would try and show that whenever the function f has a local minimum, x', then f(x') is always positive. Then it follows the function is always positive.

Answer 3

nt clr

Answer 4

:(

Answer 5

This helps qualitatively
http://www.wolframalpha.com/input/?i=cos%28sin+x%29-sin%28cos+x%29+

You see that the graph of the function always lies above the x axis. We now need to prove that analytically. The procedure I laid out above will do that. Or have you not learnt calculus yet?

How old are you btw?

Answer 6

Oh, no. I see an easier way now. Much easier.

What is the range of the functions sin x and cos x?

Answer 7

Once you know that, those ranges--which are identical--are the domains of cos(sin x) and sin(cos x).

The question then is can the ranges of cos(sin x) and sin(cos x) even intersect? If they no intersection, then they certainly never be equal.

Answer 8

range of sinx and cos x os -1 to 1

Answer 9

intersect at theta =45

Answer 10

Yes, so what is the range of cos(sin x) and the range of sin(cos x)?

Answer 11

i am nearly 17

Answer 12

learnt calculus a bit

Answer 13

Draw a graph of cos and sin and mark out on the y axis the ranges of cos and sin for a limited domain [-1,1].

You'll see they do not intersect.