To solve the problem, we need to relate the given options to the trigonometric identity involving [tex]\(\sec 2x\)[/tex].
Recall the identity for [tex]\(\sec 2x\)[/tex]:
[tex]\[ \sec 2x = \frac{1}{\cos 2x} \][/tex]
We need to express [tex]\(\cos 2x\)[/tex] in several ways using standard trigonometric identities and then check which options match the resultant expressions when substituted into the identity for [tex]\(\sec 2x\)[/tex].
1. First expression for [tex]\(\cos 2x\)[/tex]:
[tex]\[ \cos 2x = \cos^2 x - \sin^2 x \][/tex]
So,
[tex]\[ \sec 2x = \frac{1}{\cos 2x} = \frac{1}{\cos^2 x - \sin^2 x} \][/tex]
This matches option A.
2. Second expression for [tex]\(\cos 2x\)[/tex]:
[tex]\[ \cos 2x = 1 - 2\sin^2 x \][/tex]
So,
[tex]\[ \sec 2x = \frac{1}{\cos 2x} = \frac{1}{1 - 2\sin^2 x} \][/tex]
This matches option B.
3. Third expression for [tex]\(\cos 2x\)[/tex]:
[tex]\[ \cos 2x = 2\cos^2 x - 1 \][/tex]
So,
[tex]\[ \sec 2x = \frac{1}{\cos 2x} = \frac{1}{2\cos^2 x - 1} \][/tex]
This form isn't exactly given in the options, so it doesn't match any of them directly.
4. Check invalid options:
- Option C: [tex]\(\frac{1}{2\sin^2 x - 1}\)[/tex] does not match our forms of [tex]\(\cos 2x\)[/tex].
- Option D: [tex]\(\frac{1}{\sin^2 x - \cos^2 x}\)[/tex] is just the negative of option A, so it does not match [tex]\(\cos 2x\)[/tex].
Therefore, the correct answers are A and B:
- A. [tex]\(\frac{1}{\cos^2 x - \sin^2 x}\)[/tex]
- B. [tex]\(\frac{1}{1 - 2\sin^2 x}\)[/tex]
