Which of the following is a correct Pythagorean trigonometric identity?

• A. ${\sin }^2\theta -{\cos }^2\theta =0$
• B. ${\sin }^2\theta +{\tan }^2\theta =1$
• C. ${\tan }^2\theta +{\cot }^2\theta =1$
• D. ${\sin }^2\theta +{\cos }^2\theta =1$

Answer: (D)

Verified step by step guidance

1. Recall the fundamental Pythagorean trigonometric identity, which states that for any angle \( \theta \), the sum of the squares of sine and cosine equals 1: \[ \sin^{2}(\theta) + \cos^{2}(\theta) = 1 \]
2. Examine each given identity and compare it to the fundamental identity. The first option is \( \sin^{2}(\theta) - \cos^{2}(\theta) = 0 \), which differs by having a minus sign instead of a plus sign.
3. The second option is \( \sin^{2}(\theta) + \tan^{2}(\theta) = 1 \), which mixes sine squared and tangent squared, not matching the fundamental identity.
4. The third option is \( \tan^{2}(\theta) + \cot^{2}(\theta) = 1 \), involving tangent and cotangent, which is not a standard Pythagorean identity.
5. Therefore, the correct Pythagorean trigonometric identity is the one that matches the fundamental form: \[ \sin^{2}(\theta) + \cos^{2}(\theta) = 1 \]