Question 107071
{{{sqrt(1+x^2)/x=sqrt(1+(1/x^2))}}}

THIS IS INTERESTING!!! Thanks! I love this type of problem.

Note that the equation above is the same as

{{{sqrt(1+x^2)/x=sqrt(1+(1/x^2))/(1)}}}

By cross multiplication,

{{{x(sqrt(1+(1/x^2)))=1(sqrt(1+x^2))}}}
{{{x(sqrt(1+(1/x^2)))=sqrt(1+x^2)}}}

If you square both sides,

{{{(x(sqrt(1+(1/x^2))))^2=(sqrt(1+x^2))^2}}}
{{{x^2(1+1/x^2))=(1+x^2)}}}={{{1+x^2}}}
{{{x^2(1) + x^2(1/x^2)=1+x^2}}}
{{{x^2+ (x^2/x^2)=1+x^2}}}
{{{x^2+1=1+x^2}}}-------------we're getting close! watch the next move or
                              maybe you can do it. hint:it IS easy

The final move....

{{{x^2+1=x^2+1}}}--------------can you guess what I did? From {{{1+x^2}}}, I
                               just interchanged {{{1}}} and {{{x^2}}}


Therefore, {{{sqrt(1+x^2)/x=sqrt(1+(1/x^2))}}} is proven to be true.


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HyperBrain!