Question

 

Prove that :

`2 tan^-1 (sqrt((a-b)/(a+b))tan(x/2))=cos^-1 ((a cos x+b)/(a+b cosx))`

 

Answer 1

 

`2 tan^-1 (sqrt((a-b)/(a+b))tan(x/2))`

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

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

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

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

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

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

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

`=cos^(-1){(acosx+b)/(a+bcosx)}`