The expression that is not a variation of the Pythagorean identity is the third option.
What is the Pythagorean identity?
The Pythagorean identity can be written as:
[tex]sin^2(x) + cos^2(x) = 1[/tex]
For example, if we subtract cos^2(x) on both sides we get the second option:
[tex]sin^2(x) = 1 - cos^2(x)[/tex]
Which is a variation.
Now, let's divide both sides by cos^2(x).
[tex]sin^2(x)/cos^2(x) + cos^2(x)/cos^2(x) = 1/cos^2(x)\\\\tan^2(x) + 1 = sec^2(x)\\\\tan^2(x) - sec^2(x) = -1[/tex]
Notice that the third expression in the options looks like this one, but the one on the right side is positive. The above expression is in did a variation of the Pythagorean identity, then the one written in the options (with the 1 instead of the -1) is incorrect, meaning that it is not a variation of the Pythagorean identity.
Concluding, the correct option is the third one.
If you want to learn more about the Pythagorean identity, you can read:
https://brainly.com/question/24287773
