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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")`
Solution
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.
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