We know that tan (A+B)=sin (A+B)/cos (A+B) sin (A+B)=sin A*cos B+sin B*cos A cos (A+B)=cos A*cos B-sin A*sin B so for (x+pi/2) A=x B=pi/2 sin A=sin x sin B=1 cos A=cos x cos B=0 sin (A+B)=sin x*0+1*cos x------> cos x cos (A+B)=cos x*0-sin x*1------> -sin x tan (x+pi/2)=cos x/(-sin x)------> -cot x