Identity: 1 + cot² = csc²

This identity says that if you take the reciprocal of sine and the quotient cosine over sine, their squares always fit a Pythagorean relationship. It comes directly from the basic identity sin²θ + cos²θ = 1: divide every term by sin²θ to get 1 + cot²θ = csc²θ. It is true for every angle where sin θ is not zero, since cot θ and csc θ are undefined when sine is zero.

Use this identity when you need to simplify trigonometric expressions involving cotangent and cosecant, especially if a sine term is hidden in the problem. It is also useful in proofs, checking equivalence of expressions, and solving equations where converting everything to sine and cosine makes the algebra easier.

• Forgetting that the identity uses squares, so cot θ = 3 gives cot²θ = 9, not 3
• Using the identity when sin θ = 0, even though cot θ and csc θ are undefined there
• Confusing this with 1 + tan²θ = sec²θ, which is a different Pythagorean identity