Question

Prove that `2tan^-1(sqrt((a-b)/(a+b))tan  theta/2)=cos^-1((a costheta+b)/(a+b costheta))`

Answer 1

LHS = `2tan^-1(sqrt((a-b)/(a+b))tan  theta/2)=cos^-1{(1-(sqrt((a-b)/(a+b))tan  theta/2)^2)/(1+(sqrt((a-b)/(a+b))tan  theta/2)^2)}`     `[because2tan^-1(x)=cos^-1{(1-x^2)/(1+x^2)}]`

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

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

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

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

`=cos^-1{(a((1-tan^2  theta/2)/(1+tan^2  theta/2))+b((1+tan^2  theta/2)/(1+tan^2theta/2)))/(a((1+tan^2  theta/2)/(1+tan^2  theta/2))+b((1-tan^2  theta/2)/(1-tan^2  theta/2)))}`       `["Dividing"   N^r and D^r  by  1+tan^2  theta/2]` 

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

`=cos^-1{(acos  theta+b)/(a+bcostheta)}`=RHS