To prove tan^2x-sin^2x=tan^2xsin^2x
tan^2x-sin^2x
=(sin^2x)/(cos^2x)-sin^2x [Put tangent in terms of sine and cosine]
=(sin^2x-sin^2xcos^2x)/cos^2x [Common denominator]
=(sin^2x(1-cos^2x))/cos^2x [Factorise out the sin^2x]
=(sin^2x*sin^2x)/(cos^2x) [Since sin^2x+cos^2x=1, 1-cos^2x=sin^2x]
=tan^2xsin^2x [Since sin^2x/(cos^2)=tan^2x]