Pythagorean identities: \[ \begin{array}{l} \sin ^{2}(\theta)+\cos ^{2}(\theta)=1 \\ \tan ^{2}(\theta)+1=\sec ^{2}(\theta) \\ 1+\cot ^{2}(\theta)=\csc ^{2}(\theta) \end{array} \] Which statement(s) are true based on the Pythagorean identities? \( \sin ^{2} \theta=1+\cos ^{2} \theta \) \( 1=\sec ^{2} \theta-\tan ^{2} \theta \) \( \cot ^{2} \theta=\csc ^{2} \theta-1 \) \( 1-\tan ^{2} \theta=-\sec ^{2} \theta \) \( -\cos ^{2} \theta=\sin ^{2} \theta-1 \) DONE

We start with the known Pythagorean identities:  sin²θ + cos²θ = 1  tan²θ + 1 = sec²θ  1 + cot²θ = csc²θ Now, let’s analyze each statement one by one. 1) sin²θ = 1 + cos²θ  • From sin²θ + cos²θ = 1, we have sin²θ = 1 − cos²θ, not 1 + cos²θ.  • This statement is FALSE. 2) 1 = sec²θ − tan²θ  • Starting with tan²θ + 1 = sec²θ, rearrange to get sec²θ − tan²θ = 1.  • This statement is TRUE. 3) cot²θ = csc²θ − 1  • From 1 + cot²θ = csc²θ, rearrange to get cot²θ = csc²θ − 1.  • This statement is TRUE. 4) 1 − tan²θ = −sec²θ  • We know sec²θ = 1 + tan²θ, so −sec²θ = −(1 + tan²θ) = −1 − tan²θ.  • Thus 1 − tan²θ ≠ −1 − tan²θ (unless 1 = −1, which is false).  • This statement is FALSE. 5) −cos²θ = sin²θ − 1  • Rearranging sin²θ + cos²θ = 1 gives sin²θ − 1 = −cos²θ.  • This statement is TRUE. Therefore, the true statements are numbers 2, 3, and 5.