If tan a/2=root((1-e/1+e)) tan b/2,prove that cos b=(cos a-e) /(1-ecos a)

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Solution

Tan a/2=√(1-e)/(1+e) tan b/2
or,
tan b/2=√(1+e)/(1-e) tan a/2

squaring both sides,

tan² b/2=(1+e)/(1-e) tan² a/2
or,
(1-tan²b/2)/(1+tan²b/2)={(1-e)-(1+e)tan²a/2}/{(1-e)+(1+e)tan²a/2}
[by dividendo-componendo method]
or, cos b=(1-e-tan²a/2-etan²a/2)/(1-e+tan²a/2+etan²a/2)
or,
cos b={(1-tan²a/2)-e(1+tan²a/2)}/{(1+tan²a/2)-e(1-tan²a/2)}
or,
cosb=[{(1-tan²a/2)-e(1+tan²a/2)}/(1+tan²a/2)]/
[{(1+tan²s/2)-e(1-tan²a/2)}/(1+tan²a/2)]
or,
cos b=(cos a-e)/(1-ecos a)
[(1-tan²a/2)/(1+tan²a/2)=cos a] (Proved)

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