The best way to solve such questions is to simplify the options and see what they result in:
Simplifying 1st option:
[tex] \frac{1}{ cos^{2}x - sin^{2}x }= \frac{1}{cos(2x)}=sec(2x)
[/tex]
Simplifying 2nd option:
[tex] \frac{1}{1- 2sin^{2}x } = \frac{1}{cos(2x)}=sec(2x) [/tex]
Simplifying 3rd option:
[tex] \frac{1}{ sin^{2}x - cos^{2}x } = \frac{1}{-cos(2x)} =-sec(2x)[/tex]
Simplifying 4th option:
[tex] \frac{1}{2 sin^{2}x }= \frac{ cosec^{2}x }{2} [/tex]
So only first two option are the correct ones as they simplify to sec(2x).
