If (1-e)tan^2(β/2)=(1+e)tan^2(α/2), then prove that cosβ=(cosα-e)/(1-ecosα)

Steps to prove cosβ=(cosα-e)/(1-ecosα)

From Question

⇒(1-e)tan^2(β/2)=(1+e)tan^2(α/2)

⇒tan^2(β/2)={(1+e)/(1-e)}tan^2(α/2)

L.H.S

=cosβ

={1-tan^2(β/2)}/{1+tan^2(β/2)}

=[1-{(1+e)/(1-e)}tan^2(α/2)]/[1+{(1+e)/(1-e)}tan^2(α/2)]

=[(1-e)-{tan^2(α/2)+etan^2(α/2)}]/[(1-e)+{tan^2(α/2)+etan^2(α/2)}]

={1-tan^2(α/2)-e-etan^2(α/2)}/{1+tan^2(α/2)-e+etan^2(α/2)

Dividing 1+tan^2(α/2) both numenetor and denomenator

=[{1-tan^2(α/2)}/{1+tan^2(α/2)}-e{1+tan^2(α/2)}/{1+tan^2(α/2)}]/[{1+tan^2(α/2)}/{1+tan^2(α/2)}-e{1-tan^2(α/2)}/{1+tan^2(α/2)}]

=(cosα-e)/(1-ecosα)

=R.H.S

Hence cosβ=(cosα-e)/(1-ecosα) is proved.

Detail Information:-

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