Question

prove :

[tan theta/(1-cot theta)] +[cot theta/(1-tan theta)]=(1+tan theta +cot theta)

Answer 1

(tan[1-tan]+cot[1-cot])/({1-cot}[1-tan])

Answer 2

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

Answer 3

??

Answer 4

here i have taken x instead of theta for simplicity
LHS

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

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

= 1 / (tan(x)(1-tan(x)) + tan²(x) / (tan(x) - 1)

= 1 / (tan(x)(1-tan(x)) - tan²(x) / (1- tan(x))

= (1 - tan³(x)) / (tan(x)(1-tan(x))

= (1 - tan(x))(1 + tan(x) + tan²(x)) / (tan(x)(1-tan(x))

= (1 + tan(x) + tan²(x) / tan(x)

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

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

= RHS