Question

if sin A /sin B=m ,cos A/cos B =prove that tan A =m/n root 1-n^2/m^2 - 1​

Answer 1

Answer:

Step-by-step explanation:

Given

m=cosαcosβ  & n =cosαsinβ

hence,

(m2+n2)cos2β=((cosαcosβ)2+(cosαsinβ)2)cos2β

=(cos2αcos2β+cos2αsin2β)cos2β

=cos2α(1cos2β+1sin2β)cos2β

=cos2α(sin2β+cos2βcos2βsin2β)cos2β

=cos2α(1sin2β)

=cos2αsin2β

=(cosαsinβ)2

=n2

Continued..

=(n^2-m^2).sin^2 B. , putting n= cosA/cosB. and m=sin A/sinB.

=(cos^2 A/cos^2 B- sin^2 A/sin^2 B).sin^2 B.

={(sin^2 B.cos^2 A-cos^2 B.sin^2 A)/sin^2 B.cos^2 B}.sin^2 B.

={(1-cos^2 B).cos^2 A-cos^2 B.(1-cos^2 A)}/cos^2 B.

={cos^2 A - cos^2 A.cos^2 B-cos^2 B + cos^2 A.cos^2 B}/cos^2 B.

=( cos^2 A - cos^2 B)/cos^2 B.

= cos^2 A/cos^2 B - cos^2 B/cos^2 B.

=(cosA/cosB)^2 - 1 . , putting cosA/cosB= n

= n^2. - 1. Proved