Question

Prove the following:

`(tan5"A" - tan3"A")/(tan5"A" + tan3"A") = (sin2"A")/(sin8"A")`

Answer 1

L.H.S. = `(tan5"A" - tan3"A")/(tan5"A" + tan3"A")`

= `((sin5"A")/(cos5"A") - (sin3"A")/(cos3"A"))/((sin5"A")/(cos5"A") + (sin3"A")/(cos3"A"))`

= `(((sin5"A" cos3"A" - sin3"A" cos5"A")/(cos5"A" cos3"A")))/(((sin5"A" cos3"A" + sin3"A" cos5"A")/(cos5"A" cos3"A"))`

= `(sin5"A" cos 3"A" - cos5"A" sin3"A")/(sin5"A" cos3"A" + cos5"A" sin3"A")`

= `(sin(5"A" - 3"A"))/(sin(5"A" + 3"A"))`

= `(sin2"A")/(sin8"A")`

= R.H.S.