cos(A)=x/y. (A represents any angle ¶.)
1-tan^2(A)=1-sin^2(A)/cos^2(A)=(cos^2(A)-sin^2(A))/cos^2(A)=(2cos^2(A)-1)/cos^2(A)=
2-1/cos^2(A)=2-y^2/x^2=(2x^2-y^2)/x^2 QED.
Another way to solve it is to use cosine=(adjacent)/(hypotenuse), so x=adjacent and y=hypotenuse. Therefore, by Pythagoras, opposite=√(y^2-x^2). Tangent=(opposite)/(adjacent)=√(y^2-x^2)/x.
1-tan^2(A)=1-(y^2-x^2)/x^2=(x^2-(y^2-x^2))/x^2=(2x^2-y^2)/x^2.