Which of the following Pythagorean identities is correct?

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

Answer: (C)

Verified step by step guidance

1. Recall the fundamental Pythagorean identity in trigonometry: \(\sin^{2}x + \cos^{2}x = 1\). This identity states that the square of the sine of an angle plus the square of the cosine of the same angle always equals 1.
2. Examine each given identity and compare it to the fundamental identity. The first option is \(\sin^{2}x - \cos^{2}x = 1\), which differs by having a minus sign instead of a plus sign.
3. The second option is \(\sin^{2}x + \tan^{2}x = 1\). Since tangent is defined as \(\tan x = \frac{\sin x}{\cos x}\), this expression does not simplify to 1 directly and is not a standard Pythagorean identity.
4. The third option is \(\sin^{2}x + \cos^{2}x = 1\), which matches exactly the fundamental Pythagorean identity and is therefore correct.
5. The fourth option is \(\tan^{2}x + \cot^{2}x = 1\). Since \(\cot x = \frac{1}{\tan x}\), this expression does not simplify to 1 and is not a standard identity.