If #x =sqrt(3)/2# then prove #(sqrt(1+x) + sqrt(1-x))/x = 2# ?

1 Answer
George C.
Feb 23, 2018

See explanation...

Explanation:

Putting #x = sqrt(3)/2# we find:

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

#color(white)((sqrt(1+x)+sqrt(1-x))^2) = (1+x)+2sqrt(1-x^2)+(1-x)#

#color(white)((sqrt(1+x)+sqrt(1-x))^2) = 2+2sqrt(1-3/4)#

#color(white)((sqrt(1+x)+sqrt(1-x))^2) = 2+2sqrt(1/4)#

#color(white)((sqrt(1+x)+sqrt(1-x))^2) = 2+1#

#color(white)((sqrt(1+x)+sqrt(1-x))^2) = 3#

So:

#sqrt(1+sqrt(3)/2)+sqrt(1-sqrt(3)/2) = sqrt(3)#

and:

#(sqrt(1+sqrt(3)/2)+sqrt(1-sqrt(3)/2))/(sqrt(3)/2) = sqrt(3) -: sqrt(3)/2 = 2#

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