Question

Prove that `(1 + tan^2 A)/(1 + cot^2 A)` = sec² A – 1

Answer 1

To Prove: `((1 + tan^2 A))/((1 + cot^2 A))` = sec² A – 1

LHS.

We have, `(((1 + sin^2 A)/(cos^2 A)))/(((1 + cos^2 A)/(sin^2 A)))`

= `[(((cos^2 A + sin^2 A))/(cos^2 A))/(((sin^2 A + cos^2 A))/(sin^2 A))]`

= `((1/cos^2 A))/((1/sin^2 A))`  ...[As sin² A + cos² A = 1]

= `((sin^2 A))/((cos^2 A))`

= tan² A

= sec² A – 1

Hence, proved.