Respuesta :
Answer:
A, C, D
Step-by-step explanation:
Starting with the identity sin² θ cos² θ = 1, we can derive the following Pythagorean identities:
1. sin² θ + cos² θ = 1: This is the original identity itself. By adding sin² θ to both sides of the equation, we get sin² θ + cos² θ = sin² θ cos² θ + sin² θ, which simplifies to sin² θ + cos² θ = sin² θ (1 + cos² θ). Since sin² θ (1 + cos² θ) is equal to 1 according to the original identity, we can substitute it in and simplify to sin² θ + cos² θ = 1.
2. 1 + tan² θ = sec² θ: To derive this identity, we start with the original identity sin² θ cos² θ = 1. Dividing both sides by cos² θ, we get sin² θ = 1 / cos² θ. Since tan θ is equal to sin θ / cos θ, we can substitute sin² θ with (tan θ)² in the equation, giving us (tan θ)² = 1 / cos² θ. Rearranging the equation, we have 1 + (tan θ)² = sec² θ.
3. 1 + cot² θ = csc² θ: Similarly, starting with the original identity sin² θ cos² θ = 1, we divide both sides by sin² θ, giving us cos² θ = 1 / sin² θ. Since cot θ is equal to cos θ / sin θ, we can substitute cos² θ with (cot θ)² in the equation, resulting in (cot θ)² = 1 / sin² θ. Rearranging the equation, we have 1 + (cot θ)² = csc² θ.
To summarize, the Pythagorean identities that can be derived from the identity sin² θ cos² θ = 1 are:
A. sin² θ + cos² θ = 1
C. 1 + tan² θ = sec² θ
D. 1 + cot² θ = csc² θ
