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If tan[theta/2]=sqrt[[1-e]/[1+e]]tan[phi...

If `tan[theta/2]=sqrt[[1-e]/[1+e]]tan[phi/2]` then prove that `cosphi=[costheta-e]/[1-ecostheta]`

Text Solution

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Here, `tan[theta/2] = sqrt((1-e)/(1+e))tan(phi/2)`
we know, `tan (theta/2) = +-sqrt((1-costheta)/(1+costheta))`
So, our equation becomes,
`=>+-sqrt((1-costheta)/(1+costheta)) = sqrt((1-e)/(1+e))(+-sqrt((1-costheta)/(1+costheta)))`
`=>((1-costheta)/(1+costheta)) = ((1-e)/(1+e))((1-costheta)/(1+costheta))`
`=>(1+e)(1-costheta)(1+cosphi) = (1-e)(1+costheta)(1-cosphi)`
`=>(1+e)(1+cosphi-costheta-costhetacosphi) = (1-e)(1-cosphi+costheta-costhetacosphi)`
`=>1+cosphi-costheta-cosphicostheta+e+ecosphi-ecostheta-ecosthetacosphi = 1-cosphi+costheta-cosphicostheta-e+ecosphi-ecostheta+ecosthetacosphi`
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