Question

if tan A=n tan b
and sinA=m sin b
prove that cos square A=m square -1 upon nsquare -1

Answer 1

tan A=n tan B
implies tan B=tan A/n
             cot B=n/tan A
             cot^2 B=n^2/tan^2 A
sin A=m sin B
sin B =sin A /m
cosec B=m/sin A
 cosec^2 B=m^2/sin^2 A
we have,
cosec^2 B-cot^2 B=1
m^2/sin^2 A-n^2/tan^2 A=1
m^2/sin^2 A-n^2 cos^2A/sin^A=1
m^2-n^2 cos^2 A/sin^A=1
m^2-n^2 cos^2 A=sin^A
m^2-n^2 cos^A=1-cos^2A
m^2-1=n^2 cos^2 A- cos^2 A
m^2-1=cos^2 A(n^2-1)
m^2-1/n^2-1=cos^2 A
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