Question 602458
I'm going to use 'x' in place of 'theta' (as it's easier/faster to type out)



(cot(x) - tan(x))/(1 - tan(x)) - cot(x) = 1


(1/tan(x) - tan(x))/(1 - tan(x)) - cot(x) = 1


( 1/tan(x) - (tan(x)*tan(x))/tan(x) )/(1 - tan(x)) - cot(x) = 1


( 1/tan(x) - (tan^2(x))/tan(x) )/(1 - tan(x)) - cot(x) = 1


( ( 1-tan^2(x))/tan(x) )/(1 - tan(x)) - cot(x) = 1


( ( 1-tan^2(x))/tan(x) )/((1 - tan(x))/1) - cot(x) = 1


( ( 1-tan^2(x))/tan(x) )*(1/(1 - tan(x))) - cot(x) = 1


( 1-tan^2(x) )/(tan(x)(1 - tan(x)))  - cot(x) = 1


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


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


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


cot(x) + 1  - cot(x) = 1


(cot(x) - cot(x)) + 1 = 1


0 + 1 = 1


1 = 1


So we've shown that the identity is true.