If #x = sqrt3/2# then #{sqrt(1+x )+sqrt(1-x)}/{sqrt(1+x)- sqrt(1-x)}#?

1 Answer
Gió
Jan 6, 2015

You can start by rationalizing:
#(sqrt(1+x)+sqrt(1-x))/(sqrt(1+x)-sqrt(1-x))×(sqrt(1+x)+sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))=#

#=(sqrt(1+x)+sqrt(1-x))^2/(2x)=#

#=((1+x)+2sqrt(1+x)sqrt(1-x)+(1-x))/(2x)=#

#=(2+2sqrt(1-x^2))/(2x)=#

#=(1+sqrt(1-x^2))/(x)=#
Substituting: #x=sqrt(3)/2# you get:
#=(1+sqrt(1- 3/4))/(sqrt(3)/2)=(1+1/2)*(2/sqrt(3))=#
#=3/2*2/sqrt(3)=3/sqrt(3)=3/sqrt(3)*sqrt(3)/sqrt(3)=sqrt(3)#

Hope it is what you needed! :-)

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