Proving and applying fundamental trigonometric identities
What is the Pythagorean identity?
The fundamental Pythagorean identity relates sine and cosine
Simplify: sin²θ/(1 - cosθ)
Use sin²θ = 1 - cos²θ = (1-cosθ)(1+cosθ), then cancel (1-cosθ)
What is the double-angle formula for cos(2θ)?
All three are correct forms, but cos²θ - sin²θ is the standard form
Simplify: (1 - sin²θ)/cosθ
1 - sin²θ = cos²θ, so cos²θ/cosθ = cosθ
Verify: secθ - cosθ = sinθtanθ
secθ - cosθ = 1/cosθ - cosθ = (1-cos²θ)/cosθ = sin²θ/cosθ = sinθtanθ