If y^ 1 m + y^ -1 m = 2x , then (x^2 - 1)y₁^ 2 =

Step-by-step solution

Given y^ 1 m + y^ - 1 m = 2x Differentiating both sides with respect to x : 1 m y^ 1 m - 1 y₁ - 1 m y^ - 1 m - 1 y₁ = 2 y₁ m y ( y^ 1 m - y^ - 1 m ) = 2 y₁ ( y^ 1 m - y^ - 1 m ) = 2my Squaring both sides: y₁^2 ( y^ 1 m - y^ - 1 m )^2 = 4m^2 y^2 Using the identity (a - b)^2 = (a + b)^2 - 4ab : y₁^2 [ ( y^ 1 m + y^ - 1 m )^2 - 4 ] = 4m^2 y^2 Substituting y^ 1 m + y^ - 1 m = 2x : y₁^2 [ (2x)^2 - 4 ] = 4m^2 y^2 y₁^2 (4x^2 - 4) = 4m^2 y^2 4(x^2 - 1)y₁^2 = 4m^2 y^2 (x^2 - 1)y₁^2 = m^2 y^2 Answer: m^2 y^2