Question

Prove that `(sin "A" - 2sin^3 "A")/(2cos^3 "A" - cos "A") = tan "A"`

Answer 1

Given

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

L.H.S = `(sin "A" - 2 sin^3 "A")/(2 cos^3 "A" - cos "A")`

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

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

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

= `sin"A"/cos"A"`

= tan A

= R.H.S

Hence proved.