Question

prove that (1/tan3x)+(tanx-1/cot3x)+cotx=cot4x

Answer 1

1 / (tan(3x) - tan(x)) - 1/(cot(3x) - cot(x)) => 
1 / (sin(3x)/cos(3x) - sin(x)/cos(x)) - 1 / (cos(3x)/sin(3x) - cos(x)/sin(x)) => 
1 / ((sin(3x) * cos(x) - sin(x) * cos(3x)) / (cos(3x)cos(x))) - 1 / ((cos(3x)sin(x) - sin(3x)cos(x)) / (sin(x)sin(3x))) => 
cos(x)cos(3x) / sin(3x - x) - sin(x)sin(3x) / sin(x - 3x) => 
cos(x)cos(3x) / sin(2x) - sin(x)sin(3x) / sin(-2x) => 
cos(x)cos(3x) / sin(2x) + sin(x)sin(3x) / sin(2x) => 
(cos(3x)cos(x) + sin(3x)sin(x)) / sin(2x) => 
cos(3x - x) / sin(2x) => 
cos(2x) / sin(2x) => 
cot(2x)