prove that `2tan^-1(sqrt((a-b)/(a+b))tan(theta/2))`=`cos^-1((acostheta+b)/(a+bcostheta))`

Explanation

`2tan^-1(sqrt((a-b)/(a+b))tan(theta/2))=cos^-1(frac{1-(a-b)/(a+b)tan(theta/2)}{1-(a-b)/(a+b)tan(theta/2)})`


`=cos^(-1)(frac{a+b-(a-b)tan^2(theta/2)}{a+b+(a-b)tan^2(theta/2)})`

`=cos^(-1)(frac{a(1-tan^2(theta/2))+b(1+tan^2(theta/2))}{a+atan^2(theta/2)+b(1-tan^2(theta/2))})`

`=cos^(-1)(frac{cos^2(theta/2)}{1+b(cos^2(theta/2)-sin^2(theta/2))cos^2(theta/2)})`

`cos^(-1)frac{a cos theta+b}{a+b cos theta}`