Expert: Sherman D. - 11/13/2007

Question
Prove that Cot(Theta)Divided by (1-Tan(Theta))+Tan(Theta)Divided by (1-Cot(theta) is equal to 1+Tan(theta)+Cot(theta)
Show step by step how the left side of the equation is changed to convert to right side using identities.
Thank you

Answer
(cot(A)/(1-tan(A))) + (tan(A)/(1-cot(A)) = 1 + tan(A) + cot(A)

((cos(A)/sin(A))/(1 - (sin(A)/cos(A))) + ((sin(A)/cos(A))/(1 - (cos(A)/sin(A)))

((cos(A)/sin(A))/((cos(A) - sin(A))/cos(A))) + ((sin(A)/cos(A))/((sin(A) - cos(A))/sin(A)))

((cos(A)^2)/(sin(A)(cos(A) - sin(A))) + ((sin(A)^2)/(cos(A)(sin(A) - cos(A)))

((cos(A)^2)/(sin(A)(cos(A) - sin(A))) + ((sin(A)^2)/(cos(A)(-cos(A) + sin(A)))

((cos(A)^2)/(sin(A)(cos(A) - sin(A))) + ((sin(A)^2)/(-cos(A)(cos(A) - sin(A)))

((cos(A)^2)/(sin(A)(cos(A) - sin(A))) - ((sin(A)^2)/(cos(A)(cos(A) - sin(A)))

(cos(A)^3 - sin(A)^3)/(sin(A)cos(A)(cos(A) - sin(A)))

((cos(A) - sin(A))(cos(A)^2 + sin(A)cos(A) + sin(A)^2))/(sin(A)cos(A)(cos(A) - sin(A)))

(cos(A)^2 + sin(A)cos(A) + sin(A)^2)/(sin(A)cos(A))

((cos(A)^2 + sin(A)^2) + sin(A)cos(A))/(sin(A)cos(A))

(1 + sin(A)cos(A))/(sin(A)cos(A))

(1/(sin(A)cos(A))) + 1

------------------------------

1 + tan(A) + cot(A)
1 + (tan(A) + cot(A))
1 + ((sin(A)/cos(A)) + (cos(A)/sin(A)))
1 + ((sin(A)^2 + cos(A)^2)/(sin(A)cos(A)))
1 + (1/(sin(A)cos(A))

so

(cot(A)/(1 - tan(A))) + (tan(A)/(1 - cot(A)) = 1 + tan(A) + cot(A)