If tan theta/2= √{(1-e)/(1+e)} tan phi/2, prove that cosphi =( cos theta -e)/(1-ecos theta )

Tanθ/2=√(1-e)/(1+e) tanФ/2or, tanФ/2=√(1+e)/(1-e) tanθ/2squaring both sides, tan²Ф/2=(1+e)/(1-e) tan²θ/2or, (1-tan²Ф/2)/(1+tan²Ф/2)={(1-e)-(1+e)tan²θ/2}/{(1-e)+(1+e)tan²θ/2}[by dividendo-componendo method]or, cosФ=(1-e-tan²θ/2-etan²θ/2)/(1-e+tan²θ/2+etan²θ/2)or, cosФ={(1-tan²θ/2)-e(1+tan²θ/2)}/{(1+tan²θ/2)-e(1-tan²θ/2)}or, cosФ=[{(1-tan²θ/2)-e(1+tan²θ/2)}/(1+tan²θ/2)]/ [{(1+tan²θ/2)-e(1-tan²θ/2)}/(1+tan²θ/2)]or, cosФ=(cosθ-e)/(1-ecosθ) [∵, (1-tan²θ/2)/(1+tan²θ/2)=cosθ] (Proved)