Show that `cos^(-1) ((cos alpha+cos beta)/(1+cosalpha cosbeta))=2 tan^(-1)(tan (alpha/2) tan (beta/2))`

Explanation

`RHS=2 tan^ (-1)[tan frac{alpha}{2} tan frac{beta}{2}]`

`=cos^ (-1)[frac{1-tan ^{2} alpha / 2 tan ^{2} beta / 2}{1+tan ^{2} alpha / 2 tan ^{2} beta / 2}]`

`=cos ^{-1}[frac{frac{1-sin ^{2} alpha / 2 sin ^{2} beta / 2}{cos ^{2} alpha / 2 cos ^{2} beta / 2}}{1+frac{sin ^{2} alpha / 2 sin ^{2} beta / 2}{cos ^{2} alpha / 2 cos ^{2} beta / 2}}]`

`=cos ^{-1}[frac{cos ^{2} alpha / 2 cos ^{2} beta / 2-sin ^{2} alpha / 2 sin ^{2} beta / 2}{cos ^{2} alpha / 2 cos ^{2} beta / 2+sin ^{2} alpha / 2 sin ^{2} beta / 2}]`

`=cos ^{-1}[frac{2 cos ^{2} alpha / 2 cos ^{2} beta / 2-sin ^{2} alpha / 2 sin ^{2} beta / 2}{2 cos ^{2} alpha / 2 cos ^{2} beta / 2+2 sin ^{2} alpha / 2 sin ^{2} beta / 2}]`

`=cos ^{-1}[frac{(1+cos alpha)(1+cos beta)-(1-cos alpha)(1-cos beta)}{(1+cos alpha)(1+cos beta)+(1-cos alpha)(1-cos beta)}]`

`=cos ^{-1}[frac{cos alpha+cos beta}{1-cos alpha cos beta}]`

`=LHS`