Show the trigonometric equation has no solutions

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SUMMARY

The trigonometric equation $\sin (\cos a) = \cos (\sin a)$ has been proven to have no solutions. This conclusion is drawn from analyzing the ranges of the sine and cosine functions. Specifically, since $\sin (\cos a)$ is constrained between 0 and 1, while $\cos (\sin a)$ is constrained between 0 and 1 as well, the equality can only hold under specific conditions. However, through rigorous examination, it is established that no value of 'a' satisfies this equation.

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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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