Question

1+tan^2(theta) divided by 1+cot^2(theta) is equal to
(1-tan(theta)}^2 divided by
{1-cot(theta)}^2
plese prove it in details. Please help me

Answer 1

Reduce the left-hand-side using standard identities:
1+tan^2(theta) divided by 1+cot^2(theta)
=sec²(θ)/csc²(θ)
=sin²(θ)/cos²(θ)

The right-hand-side can be similarly reduced using:
(1-tan(theta)}^2 divided by
{1-cot(theta)}^2
=((1-tan(θ))/(1-cot(θ)))²
Reduce the tan() and cot() to sines and cosines, simplify and square to get the simplified form of the left-hand-side above.