An Elementary and Rigorous Proof of the Existence, Uniqueness, and Asymptotic Behavior of Solutions to tan x = x²

This paper provides a fully elementary, novel, and rigorous framework for understanding the solution structure of a classic transcendental equation. tan x = x², establishing solutions' existence, uniqueness in each interval and the asymptotic behavior of the solution sequence. In contrast to conventional approaches that rely on numerical methods or Taylor expansions, we employ only elementary techniques that involve precise analysis of derivative behavior, basic limit analysis, and carefully constructed interval arguments. In each interval of the form ((n-1/2) π,(n+1/2)π) where n is a positive integer, We first prove that the equation admits exactly one real solution. This is achieved by studying the monotonic structure of the function f(x) = tan x - x², using third derivative information to constrain the locations of local extrema and guarantee uniqueness. We then prove that no solution exists in the initial interval (0, π/2), via a precise comparison of the growth of sec²x and 2x over a cleverly and carefully chosen subinterval. Finally, we examine the asymptotic behavior of the solution sequence {x_n} and demonstrate, employing a similar approach to that of the existence part to confirm the asymptotic behavior.