Alright, let's analyze each of the given expressions to determine which of them are trigonometric identities.
### Expression A:
\(\tan^2 x = 1 + \sec^2 x\)
To evaluate this expression, let's use some known trigonometric identities:
[tex]\[
\tan^2 x = \sec^2 x - 1
\][/tex]
Plugging this into the left-hand side:
[tex]\[
\tan^2 x = \sec^2 x - 1
\][/tex]
Thus:
[tex]\[
\sec^2 x - 1 = 1 + \sec^2 x
\][/tex]
Here, subtracting \(\sec^2 x\) from both sides yields:
[tex]\[
-1 = 1
\][/tex]
This is a contradiction, so \( \tan^2 x \neq 1 + \sec^2 x \).
Therefore, Expression A is not an identity.
### Expression B:
\(\sin^2 x = 1 - \cos^2 x\)
Rewriting \(1 - \cos^2 x\):
[tex]\[
1 - \cos^2 x = \sin^2 x
\][/tex]
This follows directly from the Pythagorean identity:
[tex]\[
\sin^2 x + \cos^2 x = 1
\][/tex]
Subtracting \(\cos^2 x\) from both sides gives:
[tex]\[
\sin^2 x = 1 - \cos^2 x
\][/tex]
Therefore, Expression B is an identity.
### Expression C:
\(\sin^2 x - \cos^2 x = 1\)
Starting from the Pythagorean identity:
[tex]\[
\sin^2 x + \cos^2 x = 1
\][/tex]
Let's test the expression by substituting known angles. For example, let \(x = 0\):
[tex]\[
\sin^2 (0) - \cos^2 (0) = 0 - 1 = -1 \neq 1
\][/tex]
Therefore, the given equation does not hold for \(x = 0\).
Thus, Expression C is not an identity.
### Expression D:
\(\cot^2 x = \csc^2 x - 1\)
To analyze this, let's use known identities:
[tex]\[
\cot^2 x = \frac{\cos^2 x}{\sin^2 x}
\][/tex]
and
[tex]\[
\csc^2 x = \frac{1}{\sin^2 x}
\][/tex]
Rewriting the right-hand side:
[tex]\[
\csc^2 x - 1 = \frac{1}{\sin^2 x} - 1
\][/tex]
To combine into a single fraction:
[tex]\[
\frac{1}{\sin^2 x} - 1 = \frac{1 - \sin^2 x}{\sin^2 x} = \frac{\cos^2 x}{\sin^2 x} = \cot^2 x
\][/tex]
Hence:
[tex]\[
\cot^2 x = \csc^2 x - 1
\][/tex]
Therefore, Expression D is an identity.
### Conclusion
Based on our evaluations, the correct identities are:
- B. \(\sin^2 x = 1 - \cos^2 x\)
- D. \(\cot^2 x = \csc^2 x - 1\)
The identities are B and D.
