tan(1)+tan(2)+...+tan(89)= ?
angle dimension is in degrees
Solution :
tan(a+b)=[tan(a)+tan(b)]/[1-tan(a)*tan(b)]
tan(90)=tan(1+89)=[tan(1)+tan(89)]/[1-tan(1)*tan(89)]
tan(90)=sin(90)/cos(90)=1/0 is undefined
lim tan(x) as x--> 90 equals to infinity
lim [tan(1)+tan(89)]/[1-tan(1)*tan(89)] equals to infinity
this means that the denominator equals to 0
1-tan(1)*tan(89)=0
tan(1)*tan(89)=1
tan(1)+tan(89)=1
tan(2)+tan(88)=1
tan(3)+tan(87)=1
................
tan(44)+tan(46)=1
tan(45)=1
----------------- +
tan(1)+tan(2)+...+tan(89)=45
