ANSWER
sec²θ + tan²θ = 1
EXPLANATION
The main Pythagorean identity is,
[tex]\cos^2\theta+\sin^2\theta=1[/tex]
If we subtract sin²θ from both sides, we obtain the second option given,
[tex]\begin{gathered} \cos^{2}\theta+\sin^{2}\theta-\sin^{2}\theta=1-\sin^{2}\theta \\ \\ \cos^2\theta=1-\sin^2\theta \end{gathered}[/tex]
And, if we divide both sides by cos²θ, we obtain the third option given,
[tex]\begin{gathered} \frac{\cos^2\theta+\sin^2\theta}{\cos^2\theta}=\frac{1}{\cos^2\theta} \\ \\ 1+\tan^2\theta=\sec^2\theta \end{gathered}[/tex]
Hence, the last option, sec²θ + tan²θ = 1 is not a valid Pythagorean identity.
