Question 293344
{{{ (sqrt(6)-sqrt(2))/4 =  sqrt(2-sqrt(3))/2 }}} Start with the given equation.



{{{ sqrt(6)-sqrt(2) =  2*sqrt(2-sqrt(3)) }}} Multiply both sides by the LCD 4 to clear out the fractions.



{{{ sqrt(6)-sqrt(2) =  sqrt(4)*sqrt(2-sqrt(3)) }}} Rewrite {{{2}}} as {{{sqrt(4)}}}



{{{ sqrt(6)-sqrt(2) =  sqrt(4(2-sqrt(3))) }}} Combine the roots on the right side using the identity {{{sqrt(x)*sqrt(y)=sqrt(x*y)}}}



{{{ sqrt(6)-sqrt(2) =  sqrt(8-4*sqrt(3)) }}} Distribute and multiply



{{{sqrt(8-4*sqrt(3)) = sqrt(6)-sqrt(2) }}} Rearrange the equation.



We essentially have a nested radical on the left side. In general, we have an equation in the form {{{sqrt(a+b*sqrt(c))=sqrt(d)+sqrt(e)}}}. The goal from here is to square both sides, gather like terms, and equate rational and irrational parts. So let's do these next tasks. 



{{{sqrt(8-4*sqrt(3)) = sqrt(6)-sqrt(2) }}} Start with the given equation.



{{{8-4*sqrt(3) = (sqrt(6)-sqrt(2))^2 }}} Square both sides.



{{{8-4*sqrt(3) = (sqrt(6))^2-2*sqrt(6)*sqrt(2)+(sqrt(2))^2 }}} FOIL



{{{8-4*sqrt(3) = 6-2*sqrt(6)*sqrt(2)+(sqrt(2))^2 }}} Square {{{sqrt(6)}}} to get 6.



{{{8-4*sqrt(3) = 6-2*sqrt(6)*sqrt(2)+2 }}} Square {{{sqrt(2)}}} to get 2.



{{{8-4*sqrt(3) = 8-2*sqrt(6)*sqrt(2) }}} Combine like terms.



{{{8-4*sqrt(3) = 8-2*sqrt(6*2) }}} Combine the roots on the right side using the identity {{{sqrt(x)*sqrt(y)=sqrt(x*y)}}}



{{{8-4*sqrt(3) = 8-2*sqrt(12) }}} Multiply



{{{8-4*sqrt(3) = 8-2*sqrt(4*3) }}} Factor 12 into 4*3 (take note that 4 is a perfect square)



{{{8-4*sqrt(3) = 8-2*sqrt(4)*sqrt(3) }}} Break up the root using the identity given above.



{{{8-4*sqrt(3) = 8-2*2*sqrt(3) }}} Take the square root of 4 to get 2.



{{{8-4*sqrt(3) = 8-4*sqrt(3) }}} Multiply



From here, we can see that the two numbers are equal since the rational parts are equal and the irrational parts are equal. So this confirms that {{{ (sqrt(6)-sqrt(2))/4 =  sqrt(2-sqrt(3))/2 }}}



Note: Ideally, you'll only manipulate one side to transform it into the other. However, this is much more difficult to do with nested radicals. I find that algebraic manipulations suffice in this case. As a final check, you can use a calculator to find that both sides approximate to the value 0.258819045 (which helps verify the answer).