What is the relationship defined by the Pythagorean identity for cosecant and cotangent?

What is the relationship defined by the Pythagorean identity for cosecant and cotangent?

• A. csc^2 + cot^2 = 1
• B. csc^2 - cot^2 = 1
• C. csc^2 * cot^2 = 1
• D. csc^2 / cot^2 = 1

Answer: (B)

The Pythagorean identity that relates cosecant and cotangent is correctly expressed as csc²(θ) - cot²(θ) = 1. This relation is derived from the fundamental Pythagorean identity in trigonometry, which states that sin²(θ) + cos²(θ) = 1. Cosecant and cotangent are defined in terms of sine and cosine as follows: - Cosecant is the reciprocal of sine, csc(θ) = 1/sin(θ). - Cotangent is the quotient of cosine and sine, cot(θ) = cos(θ)/sin(θ). When you square both cosecant and cotangent, you obtain: csc²(θ) = 1/sin²(θ) and cot²(θ) = cos²(θ)/sin²(θ). Substituting these into the identity gives: 1/sin²(θ) - cos²(θ)/sin²(θ) = 1. Combining terms yields: (1 - cos²(θ))/sin²(θ) = 1. Using the identity 1 - cos²(θ) = sin²(θ) leads

The Pythagorean identity that relates cosecant and cotangent is correctly expressed as csc²(θ) - cot²(θ) = 1. This relation is derived from the fundamental Pythagorean identity in trigonometry, which states that sin²(θ) + cos²(θ) = 1.

Cosecant and cotangent are defined in terms of sine and cosine as follows:

• Cosecant is the reciprocal of sine, csc(θ) = 1/sin(θ).

• Cotangent is the quotient of cosine and sine, cot(θ) = cos(θ)/sin(θ).

When you square both cosecant and cotangent, you obtain:

csc²(θ) = 1/sin²(θ) and cot²(θ) = cos²(θ)/sin²(θ).

Substituting these into the identity gives:

1/sin²(θ) - cos²(θ)/sin²(θ) = 1.

Combining terms yields:

(1 - cos²(θ))/sin²(θ) = 1.

Using the identity 1 - cos²(θ) = sin²(θ) leads