Verified answer

recordemos que:

tg (A-B)= (tg(A)-tg(B)) / (1+tg(A).tg(B))

ademas:

tg(arctg(A))=A

arctg(A)-arctg(B)=arctg(C).....ec1

sacamos la tg a ambos miembros de la ecuacion:

tg(arctg(A)-arctg(B))=tg(arctg(C))

(tg( arctg(A) )-tg( arctg(B) )) / (1+tg( arctg(A) ).tg( arctg(B) )) = C

(A-B)/(1+A.B) = C

entonces reemplazando en la ec1:

arctg(A)-arctg(B)=arctg((A-B)/(1+A.B))

para tu ejemplo: A=2/Y y B=1/2

A-B= 2/Y-1/2 = (4-Y)/2Y

1+AB= 1+(2/Y)(1/2)= 1+(1/Y) = (Y+1)/Y

(A-B)/(1+AB) = (4-Y)/(2Y+2)

arctg(2/Y)-arctg(1/2)=arctg( (4-Y)/(2Y+2) )

usando la expresion:

arctan(A) - arctan(B) = arcta((A-B)/(1+AB))

tomando A =2/y y B= 1/2, tenemos:

arctan(2/y) - arctan(1/2) = arcta((2/y - 1/2)/(1+(2/y)(1/2)))

simplificando:

arctan(2/y) - arctan(1/2) = arcta(((4-y)/2y)/(1+1/y))

arctan(2/y) - arctan(1/2) = arcta(((4-y)/2y)/((y+1)/y)))

arctan(2/y) - arctan(1/2) = arcta(((4-y)/2(y +1)))

Y e.e.r