Question

how do i prove 2sec^2x-2sec^2xsin^2-sin^2x-cos^2x=1

Answer 1

$2\sec^2x-2\sec^2x\sin^2x-\sin^2x-\cos^2x=2\sec^2x(1-\sin^2x)-(\sin^2x+\cos^2x)$

Since $\sin^2x+\cos^2x=1$, you have

$2\sec^2x(1-\sin^2x)-(\sin^2x+\cos^2x)=2\sec^2x\cos^2x-1$

and since $\sec x=\dfrac1{\cos x}$, you end up with $2-1=1$.