Question

1+tanA*tanA/1+cotA*cotA=1-tanA/1-cotA

Prove it

Answer 1

Both LHS and RHS are separately simplified here.
LHS:
(1+tan²A)/(1+cot²A)
=(1+tan²A)/(1+1/tan²A)
=(1+tan²A)/[(1+tan²A)/tan²A]
=tan²A

RHS:
(1-tanA)²/(1-cotA)²
=(1-2tanA+tan²A)/(1-2cotA+cot²A)
=(1-2tanA+tan²A)/(1-2/tanA+1/tan²A)
=(1-2tanA+tan²A)/[(tan²A-2tanA+1)/tan²A]
=tan²A
∴, LHS=RHS(Proved)

Hope this helps.
good luck.