Question

证明当x≥1时arctan 根号(x^2-1)+arcsin1/x=π/2

Answer 1

设arctan 根号(x^2-1)=a

arcsin1/x=b

则sina=根号(x^2-1)/x cosa=1/x

sinb=1/x cosb=根号(x^2-1)/x

sin(a+b)=sinacosb+cosasinb=根号(x^2-1)/x *根号(x^2-1)/x+1/x*1/x=1=sinπ/2

(a+b)=π/2