当X≥1时,arctanx+arccox(2x/1+x2)=π/4

题目

当X≥1时,arctanx+arccox(2x/1+x2)=π/4

答案

原式是:arctanx+arccos(2x/1+x2)=π/4?①
sinarctanx=x/(√(1+x^2));cosarctanx=1/(√(1+x^2));
cosarccos(2x/1+x2)=2x/(1+x^2);sinarccos(2x/1+x2)=(1-x^2)/(1+x^2);
①式两边求sin得:x/(√(1+x^2))*2x/(1+x^2)+1/(√(1+x^2))*(1-x^2)/(1+x^2)=√2/2;
即2x^2+1-x^2=√(1+x^2)*(1+x^2)*√2/2;√(1+x^2)=√2;
X=1