Question

Prove the following:

`(cot"A"cot4"A" + 1)/(cot"A" cot4"A" - 1) = (cos3"A")/(cos5"A")`

Answer 1

L.H.S. = `(cot"A"cot4"A" + 1)/(cot"A" cot4"A" - 1)`

= `(cos"A"/(sin"A")*(cos4"A")/(sin4"A") + 1)/(cos"A"/(sin"A")*(cos4"A")/(sin4"A") - 1`

= `((cos"A"cos4"A"+sin"A"sin4"A")/(sin"A"sin4"A"))/((cos"A"cos4"A"-sin"A"sin4"A")/(sin"A"sin4"A"))`

= `(cos4"A" cos"A" + sin4"A" sin"A")/(cos4"A" cos"A" - sin4"A" sin"A")`

= `(cos(4"A" - "A"))/(cos(4"A" + "A"))`

= `(cos3"A")/(cos5"A")`

= R.H.S.