this was in my homework can anyone solve this

Question

(x + sqrt(x² -1))² + (x + sqrt(x² -1))^-2 + 2(1-2x²)

simplify this using identities

Answer 1

TO simplify the notations let M = sqrt(x^2-1)

Then M^2 = x^2-1

(x + M)^2 + (X + M)^(-2) + 2(1 - 2x^2) =

(x^2 + 2xM + M^2 + 2 - 4x^2 ) + (X + M)^(-2) =

(x^2 + 2xM + x^2-1 + 2 - 4x^2) + (x + M)^(-2) =

(2xM - 2x^2 + 1) + (x+M)^(-2) =

[ (2xM - 2x^2 + 1)(x+M)^2 + 1 ]/(x+M)^2