8-4

Section 8-4 is all about substation. The key to this section is memorizing identities and remembering some algebra. If you aren’t good at memorizing you can derive the identities and just remember 3 of them. If you can memorize them these are the ones you should know:

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 following guidelines will help you solve some of these equations:
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.
Example Problems:

Example 1:
1-cos^2x cos^2x = tan^2x
sin^2x/cos^2x = tan^2x

Example 2:
sec^2x – 1(csc^2x – 1) = 1
tan^2x(cot^2x)
sin^2x/cos^2x(cos^2x/sin^2x) = 1
(Cancel each other out)