Question

证明恒等式arctanx+arccotx=π/2 , f(x) = arctan...

证明恒等式arctanx+arccotx=π/2 , f(x) = arctanx+arccotx, 则有f'(x) = 1/(1 + x^2) - 1/(1 + x^2) = 0, f(x) = arctanx+arccotx, 则有f'(x) = 1/(1 + x^2) - 1/(1 + x^2) = 0, 所以由那个定理,f(x)是常数.把x = 1代入,得到 f(1) = arctan 1 + arccot 1 = π/2 所以f(x) = arctanx + arccotx = π/2 这个题的答案为什么要令f(1)=π/2?可以随便假设一个常数?

Answer 1

那个f'(x)就相当于导数,倒数为零就意味着f(x)的图像为一条水平线,即f(x)为一常数,所以无论是谁都得TT/2