Prove the following identities:

Question

1. (tan theta - sin theta)^2 + (1-cos theta)^2 = (1-sec theta) ^2
2. (1-2cos^2 theta) / sin theta cos theta = tan theta - cot theta
3. (sin theta + cos theta ) ^2 + (sin theta - cos theta ) ^2 = 2

Thank you so much! :)

Answer 1

sinθ = tanθ cosθ, so
tanθ - sinθ = tanθ (1-cosθ)
squared that is tan^2θ (1-cosθ)^2

now we can add up the left side:

tan^2θ(1-cosθ)^2 + (1-cosθ)^2
= (tan^2θ + 1)(1-cosθ)^2
= sec^2θ (1-cosθ)^2

on the right side, we have

1-secθ = (cosθ-1)/cosθ
squared, that is sec^2θ (cosθ-1)^2

and they are the same