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

1.	Prove that `2 tan^(-1) sqrt(b/a) =cos^(-1) ((a-b)/(a+b))`
Answer» LHS=`2tan^(-1)sqrt(b/a)` Let `tan^(-1)sqrt(b/a)=theta` `sqrt(b/a)=tantheta` `b/a=tan^2theta` `2theta` `cos2theta=(1-tan^2theta)/(1+tan^2theta)` `2theta=cos^(-1)((1-tan^2theta)/(1+tan^2theta))` `cos^(-1)((1-(b/a))/(1+(b/a)))` `cos^(-11)((a-b)/(a+b))`=RHS

Discussion

`tan^(-1)[(cosx)/(1+sinx)]`is equal to`pi/4-x/2,forx in (-pi/2,(3pi)/2)``pi/4-x/2,forx in (-pi/2,pi/2)``pi/4-x/2,forx in (-pi/2,pi/2)``pi/4-x/2,forx in (-(3pi)/2,pi/2)`
The value of `sin^(-1)(sin12)+sin^(-1)(cos12)=`
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