Which of the following is a correct Pythagorean identity in trigonometry?

• A. ${\sin }^2\theta +{\cos }^2\theta =1$
• 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: (A)

Verified step by step guidance

1. Recall the fundamental Pythagorean identity in trigonometry, which relates the squares of sine and cosine functions for the same angle \( \theta \). This identity is derived from the Pythagorean theorem applied to the unit circle.
2. Write down the standard Pythagorean identity: \( \sin^{2}(\theta) + \cos^{2}(\theta) = 1 \)
3. Examine each given option and compare it to the standard identity. The first option matches the form \( \sin^{2}(\theta) + \cos^{2}(\theta) = 1 \), which is the well-known Pythagorean identity.
4. Understand why the other options are incorrect: - \( \sin^{2}(\theta) + \tan^{2}(\theta) = 1 \) is not a standard identity. - \( \tan^{2}(\theta) + \cot^{2}(\theta) = 1 \) is also not true. - \( \sin^{2}(\theta) - \cos^{2}(\theta) = 1 \) contradicts the Pythagorean identity.
5. Conclude that the correct Pythagorean identity is the one that states \( \sin^{2}(\theta) + \cos^{2}(\theta) = 1 \), which is fundamental in trigonometry and holds for all angles \( \theta \).