If $$x^2=y+z,y^2=z+x,z^2=x+y$$​, then the value of $$\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}$$​ will be:

Correct option is A

Given:

​$$x^2 = y + z$$​​

​$$y^2 = z + x$$​​

​$$z^2 = x + y$$​​

We are asked to find the value of:

​$$\frac{1}{z+1} + \frac{1}{x+1} + \frac{1}{y+1}$$​​

Solution:

​$$x^2 = y + z$$     ...(1)

​$$y^2 = z + x$$    ....(2)

​$$z^2 = x + y$$    ...(3)

From equation (1) , we add x on both sides:

x² + x = x + y + z 

x (x + 1) = x + y + z 

​$$\frac{1}{x+1} =\frac{x}{x+y+z}$$   ...(4)

Similarly, From equation (2) , we add y on both sides:

​$$\frac{1}{y+1} =\frac{y}{x+y+z}$$      ...(5)

​Similarly, From equation (3) , we add z on both sides:

​$$\frac{1}{z+1} =\frac{z}{x+y+z}$$      ....(6)

Adding equation 4, 5 and 6 , we have-

​$$\frac{1}{x+1} + \frac{1}{y+1}+\frac{1}{z+1} =$$$$\frac{x}{x+y+z} +\frac{y}{x+y+z}+\frac{z}{x+y+z}$$​​​

=$$\frac{x+y+z}{x+y+z}$$ = 1​