Question

Prove the following:

`1/(tan3"A" - tan"A") - 1/(cot3"A" - cot"A")` = cot2A

Answer 1

L.H.S. = `1/(tan3"A" - tan"A") - 1/(cot3"A" - cot"A")`

= `1/(tan3"A" - tan"A") - 1/(1/(tan3"A") - 1/tan"A"`

= `1/(tan3"A" - tan"A") - (tan3"A"*tan"A")/(tan"A" - tan3"A")`

= `1/(tan3"A" - tan"A") + (tan3"A"*tan"A")/(tan3"A" - tan"A")`

= `(1 + tan3"A"*tan"A")/(tan3"A" - tan"A")`

= `1/(((tan3"A" - tan"A")/(1 + tan3"A"*tan"A"))`

= `1/(tan(3"A" - "A"))`

= `1/(tan2"A")`

= cot 2A

= R.H.S.