To determine which equation is true based on Pythagorean identities, let's analyze each option step by step.
### Option 1:
[tex]\(\sin^2 \theta - 1 = \cos^2 \theta\)[/tex]
We know from the basic Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Let's manipulate this identity to see if it matches the given equation:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
[tex]\[
\implies \sin^2 \theta = 1 - \cos^2 \theta
\][/tex]
Now, if we subtract 1 from both sides:
[tex]\[
\sin^2 \theta - 1 = (1 - \cos^2 \theta) - 1
\][/tex]
[tex]\[
\implies \sin^2 \theta - 1 = -\cos^2 \theta
\][/tex]
This is not equal to [tex]\(\cos^2 \theta\)[/tex]. Hence, Option 1 is not true.
### Option 2:
[tex]\(\sec^2 \theta - \tan^2 \theta = -1\)[/tex]
We know the Pythagorean identity related to secant and tangent:
[tex]\[
\sec^2 \theta = 1 + \tan^2 \theta
\][/tex]
Let's rearrange this identity:
[tex]\[
\sec^2 \theta - \tan^2 \theta = 1
\][/tex]
This shows that the correct identity is [tex]\(\sec^2 \theta - \tan^2 \theta = 1\)[/tex], not [tex]\(-1\)[/tex]. Therefore, Option 2 is not true.
### Option 3:
[tex]\(-\cos^2 \theta - 1 = -\sin^2 \theta\)[/tex]
We start again with the basic Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
If we subtract 1 from both sides:
[tex]\[
-\cos^2 \theta - 1 = -(\cos^2 \theta + 1)
\][/tex]
[tex]\[
\implies -\cos^2 \theta - 1 = -\sin^2 \theta - 1
\][/tex]
Clearly, this is not equal to [tex]\(-\sin^2 \theta\)[/tex]. Therefore, Option 3 is not true.
### Option 4:
[tex]\(\cot^2 \theta - \csc^2 \theta = -1\)[/tex]
We know the Pythagorean identity related to cotangent and cosecant:
[tex]\[
\csc^2 \theta = 1 + \cot^2 \theta
\][/tex]
Rearranging this identity:
[tex]\[
\cot^2 \theta - \csc^2 \theta = -1
\][/tex]
This matches the given equation exactly. Thus, Option 4 is true.
Hence, the equation that is true based on Pythagorean identities is:
[tex]\[
\cot^2 \theta - \csc^2 \theta = -1
\][/tex]
The correct choice is: 4
