Question

Prove that `(tan^2"A")/(tan^2 "A"-1) + (cosec^2"A")/(sec^2"A"-cosec^2"A") = (1)/(1-2 co^2 "A")`

Answer 1

Taking L.H.S.

`(tan^2"A")/(tan^2 "A"-1) + (cosec^2"A")/(sec^2"A"-cosec^2"A")`

`((sin^2"A")/(cos^2"A"))/((sin^2"A")/(cos^2"A")-1)+ ((1)/(sin^2"A"))/((1)/(cos^2"A")-(1)/(sin^2"A"))  ...(∵ tan "A" = (sin"A")/(cos"A"))`

 

= `(sin^2"A")/(sin^2 "A"- cos^2"A") + (1)/(sin^2 "A"). (sin^2"A" cos^2"A")/(sin^2"A"-cos^2"A")`

= `(sin^2"A")/(sin^2 "A"- cos^2"A") + (cos^2"A")/(sin^2 "A"- cos^2"A")`

= `(sin^2 "A"+ cos^2"A")/(sin^2"A"-cos^2"A")`

= `(1)/(1-cos^2"A"-cos^2"A")   ...(∵ sin^2 "A" = 1 -cos^2"A")`

= `(1)/(1-2 cos^2 "A")`

= R.H.S.