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

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Answer 1 · 2018-03-17T16:32:15

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#