1. The problem is to identify the correct Pythagorean identities from the given options. 2. The fundamental Pythagorean identity is: $$\sin^2 x + \cos^2 x = 1$$ This identity is derived from the Pythagorean theorem applied to a right triangle on the unit circle. 3. From this identity, we can rearrange terms to get: $$\sin^2 x = 1 - \cos^2 x$$ and $$\cos^2 x = 1 - \sin^2 x$$ 4. Another set of Pythagorean identities involve tangent and secant, and cotangent and cosecant: $$1 + \tan^2 x = \sec^2 x$$ and $$1 + \cot^2 x = \csc^2 x$$ Rearranging the second gives: $$\cot^2 x - \csc^2 x = -1$$ 5. Now, let's check the given options: - $- \text{scc}^2 x = -1 - \tan^2 x$ is incorrect because the function should be $\sec$ not $\text{scc}$. - $\sin^2 + \cos^2 = 1$ is correct (assuming $\sin^2 x + \cos^2 x = 1$). - $\cot^2 x - \csc^2 x = -1$ is correct. - $\cot^2 x - \csc^2 x = 1$ is incorrect. - $\sin^2 x = 1 - \cos^2 x$ is correct. - $\text{scc}^2 x - 1 = \tan^2 x$ is incorrect due to typo; it should be $\sec^2 x - 1 = \tan^2 x$. - $- \cos^2 x = \sin^2 x + 1$ is incorrect. - $- \sin^2 x = 1 - \cos^2 x$ is incorrect. Final correct identities from the list are: $$\sin^2 x + \cos^2 x = 1$$ $$\cot^2 x - \csc^2 x = -1$$ $$\sin^2 x = 1 - \cos^2 x$$