Let's evaluate each statement against the given Pythagorean identities:
### Statement 1: [tex]\(\sin^2(\theta) = 1 + \cos^2(\theta)\)[/tex]
Using the identity:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
we can rearrange this to find [tex]\(\sin^2(\theta)\)[/tex]:
[tex]\[
\sin^2(\theta) = 1 - \cos^2(\theta)
\][/tex]
Clearly, [tex]\(\sin^2(\theta) = 1 - \cos^2(\theta)\)[/tex] is not the same as [tex]\(\sin^2(\theta) = 1 + \cos^2(\theta)\)[/tex]. Therefore, this statement is false.
### Statement 2: [tex]\(1 = \sec^2(9) - \tan^2(\theta)\)[/tex]
Using the identity:
[tex]\[
\tan^2(\theta) + 1 = \sec^2(\theta)
\][/tex]
we can rearrange this to find:
[tex]\[
\sec^2(\theta) - \tan^2(\theta) = 1
\][/tex]
However, here [tex]\(\sec^2(9)\)[/tex] is given with a specific value of [tex]\(9\)[/tex], whereas [tex]\(\tan^2(\theta)\)[/tex] uses a different [tex]\(\theta\)[/tex]. Due to the inconsistency in applying different angles, this statement cannot be verified as true. Thus, this statement is false.
### Statement 3: [tex]\(\cot^2(\theta) = \csc^2(\theta) - 1\)[/tex]
Using the identity:
[tex]\[
1 + \cot^2(\theta) = \csc^2(\theta)
\][/tex]
we can rearrange this to obtain:
[tex]\[
\cot^2(\theta) = \csc^2(\theta) - 1
\][/tex]
This matches exactly with our original statement. Therefore, this statement is true.
### Statement 4: [tex]\(1 - \tan^2(\theta) = -\sec^2(\theta)\)[/tex]
Using the identity:
[tex]\[
\tan^2(\theta) + 1 = \sec^2(\theta)
\][/tex]
we can rearrange this to find:
[tex]\[
1 - \tan^2(\theta) = -( \sec^2(\theta) - 2) \neq -\sec^2(\theta)
\][/tex]
This indicates that [tex]\(1 - \tan^2(\theta) \neq -\sec^2(\theta)\)[/tex]. Therefore, this statement is false.
### Statement 5: [tex]\(-\cos^2(\theta) = \sin^2(\theta) - 1\)[/tex]
Using the identity:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
we can rearrange this to find [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[
cos^2(\theta) = 1 - \sin^2(\theta)
\][/tex]
Multiplying through by -1, we get:
[tex]\[
-\cos^2(\theta) = \sin^2(\theta) - 1
\][/tex]
This matches exactly with our original statement. Therefore, this statement is true.
### Summary
Based on the Pythagorean identities, the evaluation of the statements are as follows:
1. False
2. False
3. True
4. False
5. True
Thus, the correct results are:
[tex]\[
\begin{array}{l}
\text{Statement 1: False} \\
\text{Statement 2: False} \\
\text{Statement 3: True} \\
\text{Statement 4: False} \\
\text{Statement 5: True} \\
\end{array}
\][/tex]
