Question
Prove the identity cos theta - sin theta / cos theta + sin theta = sec 2 theta - tan 2 theta.
Answers
Steve
omitting all the θ's for simplicity,
(cos-sin)/(cos+sin)
multiply top and bottom by (cos-sin)
(cos-sin)^2 / (cos^2 - sin^2)
(cos^2 - 2sin*cos + sin^2)/cos2θ
(1 - 2sin*cos)/cos2θ
(1-sin2θ)/cos2θ
sec2θ - tan2θ
(cos-sin)/(cos+sin)
multiply top and bottom by (cos-sin)
(cos-sin)^2 / (cos^2 - sin^2)
(cos^2 - 2sin*cos + sin^2)/cos2θ
(1 - 2sin*cos)/cos2θ
(1-sin2θ)/cos2θ
sec2θ - tan2θ