Based on Pythagorean identities, which equation is true?

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

Answer: (B)

Verified step by step guidance

1. Recall the fundamental Pythagorean identity in trigonometry, which states that for any angle \(x\), the sum of the squares of sine and cosine equals 1: \(\sin^{2}x + \cos^{2}x = 1\).
2. Examine the given equations and identify which one matches the Pythagorean identity. Notice that the identity involves the sum of \(\sin^{2}x\) and \(\cos^{2}x\), not their difference.
3. Understand that the equation \(\sin^{2}x - \cos^{2}x = 1\) is not true because subtracting \(\cos^{2}x\) from \(\sin^{2}x\) does not yield 1 for all \(x\).
4. Similarly, the equation \(\sin^{2}x - \cos^{2}x = 0\) is not the Pythagorean identity, and neither is \(\sin^{2}x + \cos^{2}x = 0\) because the sum of squares of sine and cosine cannot be zero for any real angle \(x\).
5. Therefore, the correct Pythagorean identity is the equation where the sum of \(\sin^{2}x\) and \(\cos^{2}x\) equals 1: \(\sin^{2}x + \cos^{2}x = 1\).