Final answer:
The true statement for all real values of θ, based on the Pythagorean identity, is option 3: sin²θ = 1 - cos²θ.
Step-by-step explanation:
The question asks to identify the true statement for all real values of θ. We use the Pythagorean identity in trigonometry, which states that for any angle θ, the square of the sine plus the square of the cosine equals 1:
sin2θ + cos2θ = 1
We can manipulate this identity to match the given choices:
sin2θ - cos2θ = (1 - cos2θ) - cos2θ which is not always equal to 1.
cos2θ - sin2θ = cos2θ - (1 - cos2θ) which simplifies to 2cos2θ - 1, and this is not equal to 1.
sin2θ = 1 - cos2θ is the correct rearrangement of the Pythagorean identity and thus it is true for all real values of θ.
sin2θ = cos2θ - 1 is not a correct rearrangement of the Pythagorean identity and thus not always true.
Therefore, the true statement for all real values of θ is option 3: sin2θ = 1 - cos2θ.