MHB Show the trigonometric equation has no solutions

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The trigonometric equation $\sin (\cos a) = \cos (\sin a)$ is examined for potential solutions. It is established that the range of $\sin(\cos a)$ is limited to the interval [-1, 1], while $\cos(\sin a)$ also falls within the same range. However, through analysis, it is demonstrated that these two functions do not intersect at any point within their respective ranges. Consequently, it is concluded that the equation has no solutions. The proof relies on the properties and behaviors of the sine and cosine functions.
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Show that the trigonometric equation $\sin (\cos a)= \cos (\sin a)$ has no solutions.
 
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anemone said:
Show that the trigonometric equation $\sin (\cos a)= \cos (\sin a)$ has no solutions.

if sin (x) = cos (y) then

$x+y = \pi/2 +2npi$ or $(2n+1)pi-\pi/2$

so lowest | cos a + sin a | = $\pi/2$

so |cos a + sin a| =$ \pi/2$ should have a solution

but |cos a + sin a | <= $\sqrt(2)$

so no solution
 

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