Question

Prove:
cotx/(1-tanx) + tanx/(1-cotx)= 1+tanx+cotx :)

Answer 1

= 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)