If 2 h ⁻¹ ( a 1-a^2 )= ( 1+x 1-x ) , then x is equal to

Step-by-step solution

We have given, As we know, ⁻¹ x= x+ 1+x^2 So, 2 h ⁻¹ x=2 (x+ 1+x^2 ) aligned & = (x+ 1+x^2 )^2 & = [x^2+1+x^2+2 x 1+x^2 ] & 2 h ⁻¹ x= [2 x^2+1+2 x 1+x^2 ] aligned Now, put x= a 1-a^2 aligned & = [ 2 a^2 1-a^2 +1+ 2 a 1-a^2 1 1-a^2 ] & = [ 2 a^2 1-a^2 +1+ 2 a 1-a^2 ] & = [ 2 a^2+1-a^2+2 a 1-a^2 ] & = [ a^2+2 a+1 1-a^2 ] & aligned Now, by using Eq. (i), aligned & [ a^2+2 a+1 1-a^2 ]= [ 1+x 1-x ] & a^2+2 a+1 1-a^2 = 1+x 1-x aligned [by using componendo and dividendo rule] aligned & a^2+2 a+1+1-a^2 a^2+2 a+1-1+a^2 & =