using the intermediate value theorum, prove that arctanx = arccosx has a solution.

i've gotten that the domain of arctanx is -pi/2 to pi/2 but i'm not sure where to go from here...can i move the arccosx to the left? when i do that, i can't figure f(pi/2) though..i'm confused!

1 answer

on x in [0,1]
arctan rises from 0 to pi/4

arccos falls from pi/2 to 0.

Since both functions are continuous,

arccos - arctan assumes all values between pi/2 and -pi/4

That would include 0, meaning arctan = arccos