Question

`tan^1(x+sqrt(1+x^2))`,Express the value of the above in the simplest form?

Answer 1

Let `x=tantheta=>theta=tan^-1x`

`tan^1(x+sqrt(1+x^2))`

`=tan^1(tantheta+sqrt(1+tan^2theta))`

`=tan^1(tantheta+sqrt(sec^2theta))`

`=tan^1(tantheta+sectheta)`

`=tan^1(sintheta/costheta+1/costheta)`

`=tan^1((sintheta+1)/costheta)`

`=tan^1((2sin(theta/2)cos(theta/2)+sin^2(theta/2)+cos^2(theta/2))/(cos^2(theta/2)-sin^2(theta/2)))`

`=tan^1((sin(theta/2)+cos(theta/2))^2/((cos(theta/2)-sin(theta/2))(cos(theta/2)+sin(theta/2))))`

`=tan^1((cos(theta/2)+sin(theta/2))/(cos(theta/2)-sin(theta/2)))`

`=tan^1((cos(theta/2)/cos(theta/2)+sin(theta/2)/cos(theta/2))/(cos(theta/2)/cos(theta/2)-sin(theta/2)/cos(theta/2)))`

`=tan^1((1+tan(theta/2))/(1-tan(theta/2)))`

`=tan^1((tan(pi/4)+tan(theta/2))/(1-tan(pi/4)tan(theta/2)))`

`=tan^1(tan(pi/4+theta/2))`

`=pi/4+theta/2`

`=pi/4+1/2tan^-1x`