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

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1.	If `tan[theta/2]=sqrt[[1-e]/[1+e]]tan[phi/2]` then prove that `cosphi=[costheta-e]/[1-ecostheta]`
Answer» 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` `=>2cosphi+2e = 2costheta+2ecosthetacosphi` `=>cosphi+e = costheta+ecosthetacosphi` `=>cosphi(1-ecostheta)) = costheta - e` `=>cosphi = ( costheta - e)/(1-ecostheta)`

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