Expert: Sherman D. - 3/28/2005

Question
prove the identity.

1) (tan t/1 - cot t) + (1 + cot t/tan t)= (tan^2t + 2 - csc^2t/tan t -1)

Answer
(tan(t)/(1 - cot(t)) + ((1 + cot(t))/(tan(t)) =
(tan(t)^2 + 2 - csc(t)^2)/(tan(t) - 1)

becomes

(tan(t)/(1 - cot(t)) + ((1 + cot(t))/(tan(t)) =
(tan(t)^2 + 2 - (cot(t)^2 + 1)/(tan(t) - 1)

Multiply the left side by tan(t) * (1 - cot(t))

((tan(t) * tan(t)) + ((1 - cot(t)) * (1 + cot(t)))/(tan(t) * (1 - cot(t)) =
(tan(t)^2 + 2 - cot(t)^2 - 1)/(tan(t) - 1)

(tan(t)^2) + (1 + cot(t) - cot(t) - cot(t)^2)/(tan(t) * (1 - (1/(tan(t))) =  
(tan(t)^2 + 1 - cot(t)^2)/(tan(t) - 1)

(tan(t)^2 + (1 - cot(t)^2))/(tan(t) - (tan(t)/(tan(t))) =
(tan(t)^2 + 1 - cot(t)^2)/(tan(t) - 1)

(tan(t)^2 + 1 - cot(t)^2)/(tan(t) - 1) =
(tan(t)^2 + 1 - cot(t)^2)/(tan(t) - 1)

and there you have it.

info about csc(t)^2, go to
www-physics.ucsd.edu/~aklohs/studies/math/trig.html

They will use "A" instead of "t", but its the same thing.