Proving the identity

Question

What are the steps and transformations for the expression:

(1+cos^2x) / (sin^2x) = 2csc^2x-1

For example, a part asks for 1+cos^2x = ? (and we have to choose from a tangent identity, double-angle identity, or a Pythagorean identity.

Answer 1

Just to be clear, we want to prove

(1 + cos²)/sin² = 2csc² - 1.

By definition 1/sin = csc, therefore 1/sin² = csc², so

(1 + cos²)/sin² = (1 + cos²)csc².

On the other hand, we know that sin² + cos² = 1, and so if we add 1 - sin² to both sides of this equation we get

1 + cos² = 2 - sin²,

and substituting on the previous equation we get

(1 + cos²)/sin² = (1 + cos²)csc² = (2 - sin²)csc² = 2csc² - sin²csc² = 2csc² - 1

since sin²csc² = sin² · 1/sin² = 1