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Math OLYMPIAD level problems on Trigonometry
Problem 1
Let f(x) = .
What is f^{2012}(x), where the function is being applied/composed 2012 times?
This notation indicates repeated composition of functions, not exponentiation of functions.
For example, f^2(x) = f(f(x)) and not f(x)*f(x). Similarly, f^3(x) = f(f(f(x))).
Solution
This problem allows absolutely unexpected and elegant solution.
Let x = tan(a) (which means "let a = arctan(x)"). Then
f(x) = = = .
It means that taking f(x) for x= tan(a) returns .
Obviously, that taking the composition (fof)(x) = f(f(x)) will return ;
taking the composition (fofof)(x) = f(f(f(x))) will return ;
. . . . and so on . . .
taking the composition f^{2012}(x) = f(f(f...f(x)))...) will return .
It easy to calculate: = + ; THEREFORE
= = = = = . ANSWERANSWER. f^{2012}(x) = .