Pythagorean Identities

One of the things we learned in the week was using pythagorean identities to help simplify trigonometric equations. Some identities are

1 – cos^2θ = sin^2θ
1 – sin^2θ = cos^2θ
sin^2θ – 1 = - cos^2θ
cos^2θ – 1 = - sin^2θ
csc2 x – 1 = cot^2x
sec2 x – 1 = tan^2x
sec2 x – 1 tan^2x = 1
1 – csc^2x = -cot^2x
1 – sec^2x = -tan^2x
tan^2x – sec^2x = -1
csc^2x – cot^2x = 1
cot^2 – csc^2x = -1

The steps to solving these problems are:
1) Check the problem for Pythagorean identities : sin^2θ + cos^2θ = 1
1 + tan^2θ = sec2 θ
1 + cot^2θ = csc^2θ
2) Change everything to sin, cos, or tan.
3) Use algebra to simplify the problem. (FOIL, etc)
4) Plug in anymore identities.
5) Solve the problem using algebra.


sin x (csc x - sin x)
sin x(1/sin x - sin x)
1 - sin squared x
cos squared x