Question #11b8e

#tan^-1([sqrt{1+x^2}-sqrt{1-x^2}]/ [sqrt{1+x^2}+sqrt{1-x^2}])=x#
#=>([sqrt{1+x^2}-sqrt{1-x^2}]/ [sqrt{1+x^2}+sqrt{1-x^2}])=tanx#
#=>[sqrt{1+x^2}+sqrt{1-x^2}]/ [sqrt{1+x^2}-sqrt{1-x^2}]=1/tanx#
#=>sqrt(1+x^2)/sqrt(1-x^2)=(1+tanx)/(1-tanx)# [ADDITION-DIVISION METHOD]
#=>(1+x^2)/(1-x^2)=(1+tanx)^2/(1-tanx)^2#
#=>1/x^2=((1+tanx)^2+(1-tanx)^2)/((1+tanx)^2-(1-tanx)^2#
#=>1/x^2=(2(tan^2x+1))/(4tanx#
#=>x^2=(2tanx)/(tan^2x+1)#
#=>x^2=2tanxcdot1/sec^2x#
#=>x^2=2 cdot sinx/cosxcdotcos^2x#
#=>x^2=2 cdot sinx cdot cosx#
#=>x=sqrtsin2x#

Now what I have to solve...

#tan^-1([sqrt{1+x^2}-sqrt{1-x^2}]/ [sqrt{1+x^2}+sqrt{1-x^2}])=x#

#tanx=[sqrt{1+x^2}-sqrt{1-x^2}]/ [sqrt{1+x^2}+sqrt{1-x^2}]#

#tanx*[sqrt{1+x^2}+sqrt{1-x^2}]=sqrt{1+x^2}-sqrt{1-x^2}#

#tanx*sqrt(1+x^2)+tanx*sqrt(1-x^2)=sqrt{1+x^2}-sqrt{1-x^2}#

#(1+tanx)*sqrt(1-x^2)=(1-tanx)*sqrt(1+x^2)#

#(1+tanx)^2*(1-x^2)=(1-tanx)^2*(1+x^2)#

#(1+tanx)^2-x^2*(1+tanx)^2=(1-tanx)^2+x^2*(1-tanx)^2#

#(1+tanx)^2-(1-tanx)^2=x^2*[(1+tanx)^2+(1-tanx)^2]#

#x^2*[2(tanx)^2+2]=4tanx#

#2x^2*[(tanx)^2+1]=4tanx#

#2x^2*(secx)^2=4tanx#

#x^2=(4tanx)/[2(secx)^2]#

#x^2=2tanx*(cosx)^2#

#x^2=2sinx*cosx#

#x^2=sin2x#

Hence #x=sqrt(sin2x)#