+  +  = 1.      (1)


The domain, where all included functions are defined, is the segment  [0,1].


Two obvious solutions to the given equation in this domain are x= 0  and  x= 1.


Below I will show that the given equation HAS NO other solutions.



Indeed, let  0 < x < 1.

Then         is defined and is positive number   > 0.

Similarly,   is defined and is positive number   > 0.



    For any two real positive numbers "a" and "b" the following inequality is valid

        a + b > .    (2)


    To prove it, square both sides. You will get

        a^2 + 2ab + b^2 > a^2 + b^2,

    which is valid for all positive "a" and "b".



Now apply the inequality (2) for  a=   and  b= .  You will get

     +  >  =  =  = 1.


Thus,  the sum   +   at  0 < x < 1  is just greater than 1.


With the added positive addend  ,  the sum   +  +   is just even more than 1.  


Therefore,  the sum   +  +   can not be equal to 1  at  0 < x < 1.



Thus, it is  PROVED that the given equation has no solutions inside the segment [0,1].  

So, the endpoints  x= 0  and  x= 1 are the only solutions.