Answer :
Let's analyze each of the given trigonometric expressions step-by-step to determine which ones are identities.
### Identity A: [tex]\(\tan^2 x = 1 + \sec^2 x\)[/tex]
To check if this is an identity:
- Recall the Pythagorean identity for tangent and secant: [tex]\(\tan^2 x + 1 = \sec^2 x\)[/tex].
- Subtracting 1 from both sides: [tex]\(\tan^2 x = \sec^2 x - 1\)[/tex].
Thus, [tex]\(\tan^2 x = 1 + \sec^2 x\)[/tex] does not match the known identity. Therefore, Identity A is not an identity.
### Identity B: [tex]\(\sin^2 x = 1 - \cos^2 x\)[/tex]
To check if this is an identity:
- Recall the Pythagorean identity for sine and cosine: [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex].
- Rearranging the terms: [tex]\(\sin^2 x = 1 - \cos^2 x\)[/tex].
Thus, Identity B is indeed correct. Therefore, Identity B is an identity.
### Identity C: [tex]\(\cot^2 x = \csc^2 x - 1\)[/tex]
To check if this is an identity:
- Recall the Pythagorean identity for cotangent and cosecant: [tex]\(\cot^2 x + 1 = \csc^2 x\)[/tex].
- Subtracting 1 from both sides: [tex]\(\cot^2 x = \csc^2 x - 1\)[/tex].
Thus, Identity C is indeed correct. Therefore, Identity C is an identity.
### Identity D: [tex]\(\sin^2 x - \cos^2 x = 1\)[/tex]
To check if this is an identity:
- Given the Pythagorean identities, there is no known identity that states [tex]\(\sin^2 x - \cos^2 x = 1\)[/tex].
- Rearranging [tex]\(\sin^2 x - \cos^2 x\)[/tex] does not result in 1, since the standard rearranged form of the Pythagorean identity gives: [tex]\(\sin^2 x = 1 - \cos^2 x\)[/tex] instead.
Thus, Identity D is not correct. Therefore, Identity D is not an identity.
### Summary
Based on the above analysis:
- Identity A: [tex]\(\tan^2 x = 1 + \sec^2 x\)[/tex] is not an identity.
- Identity B: [tex]\(\sin^2 x = 1 - \cos^2 x\)[/tex] is an identity.
- Identity C: [tex]\(\cot^2 x = \csc^2 x - 1\)[/tex] is an identity.
- Identity D: [tex]\(\sin^2 x - \cos^2 x = 1\)[/tex] is not an identity.
Hence, the identities among the provided options are [tex]\(\sin^2 x = 1 - \cos^2 x\)[/tex] and [tex]\(\cot^2 x = \csc^2 x - 1\)[/tex].
### Identity A: [tex]\(\tan^2 x = 1 + \sec^2 x\)[/tex]
To check if this is an identity:
- Recall the Pythagorean identity for tangent and secant: [tex]\(\tan^2 x + 1 = \sec^2 x\)[/tex].
- Subtracting 1 from both sides: [tex]\(\tan^2 x = \sec^2 x - 1\)[/tex].
Thus, [tex]\(\tan^2 x = 1 + \sec^2 x\)[/tex] does not match the known identity. Therefore, Identity A is not an identity.
### Identity B: [tex]\(\sin^2 x = 1 - \cos^2 x\)[/tex]
To check if this is an identity:
- Recall the Pythagorean identity for sine and cosine: [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex].
- Rearranging the terms: [tex]\(\sin^2 x = 1 - \cos^2 x\)[/tex].
Thus, Identity B is indeed correct. Therefore, Identity B is an identity.
### Identity C: [tex]\(\cot^2 x = \csc^2 x - 1\)[/tex]
To check if this is an identity:
- Recall the Pythagorean identity for cotangent and cosecant: [tex]\(\cot^2 x + 1 = \csc^2 x\)[/tex].
- Subtracting 1 from both sides: [tex]\(\cot^2 x = \csc^2 x - 1\)[/tex].
Thus, Identity C is indeed correct. Therefore, Identity C is an identity.
### Identity D: [tex]\(\sin^2 x - \cos^2 x = 1\)[/tex]
To check if this is an identity:
- Given the Pythagorean identities, there is no known identity that states [tex]\(\sin^2 x - \cos^2 x = 1\)[/tex].
- Rearranging [tex]\(\sin^2 x - \cos^2 x\)[/tex] does not result in 1, since the standard rearranged form of the Pythagorean identity gives: [tex]\(\sin^2 x = 1 - \cos^2 x\)[/tex] instead.
Thus, Identity D is not correct. Therefore, Identity D is not an identity.
### Summary
Based on the above analysis:
- Identity A: [tex]\(\tan^2 x = 1 + \sec^2 x\)[/tex] is not an identity.
- Identity B: [tex]\(\sin^2 x = 1 - \cos^2 x\)[/tex] is an identity.
- Identity C: [tex]\(\cot^2 x = \csc^2 x - 1\)[/tex] is an identity.
- Identity D: [tex]\(\sin^2 x - \cos^2 x = 1\)[/tex] is not an identity.
Hence, the identities among the provided options are [tex]\(\sin^2 x = 1 - \cos^2 x\)[/tex] and [tex]\(\cot^2 x = \csc^2 x - 1\)[/tex].
