Relationships Among the Functions..

These are some helpful formulas that we used:

Reciprocal Relationships:
csc(theta)=1/sin(theta)
sec(theta)=1/cos(theta)
cot(theta)=1/tan(theta)

Pythagorean Relationships:
sin^2(theta)+cos^2(theta)=1
1+tan^2(theta)=sec^2(theta)
1+cot^2(theta)=csc^2(theta)

*Steps to be loosely followed:
1. Algebra
2. Identities-try pythagorean, then move everything to sin and cos if that will help
3. Algebra
4. Continue with 1-3

Example 1:
(1-cos(theta))(1+cos(theta))
1+cos(theta)-cos(theta)-cos^2(theta)
*+cos(theta) and -cos(theta) cancel, so you are left with
1-cos^2(theta)
*You can use the formula sin^2(theta)+cos^2(theta)=1
sin^2(theta)=1-cos^2(theta)
=sin^2(theta)

Example 2:
(csc(theta)-1)(csc(theta)+1)
csc^2(theta)+csc(theta)-csc(theta)-1
*+csc(theta) and - csc(theta) cancel, so you are left with
csc^2(theta)-1
*You can use the formula 1+cot^2(theta)=csc^2(theta)
cot^2(theta)=csc^2(theta)-1
=cot^2(theta)

Example 3:
tan^2x-sec^2x
1+tan^2x=sec^2x
1=-tan^2x+sec^2x/-1
-1=tan^2x-sec^2x
=-1

Example 4:
1/sin^2(theta)-1/tan^2(theta)
1/sin^2(theta)-cos^2(theta)/sin^2(theta)
1-cos^2(theta)/sin^2(theta)=sin^2(theta)/sin^2(theta)
=1

Example 5:
1+tan^2(theta)
*this is a formula that is equal to
sec^2(theta)

Example 6:
cos^2(theta)+sin^2(theta)
*this is also another formula that is equal to
1

Example 7:
1+cot^2(theta)
*this is a formula that is equal to
csc^2(theta)

Example 8:
(secx-1)(secx+1)
sec^2x+secx-secx-1
*+secx and -secx cancel, so you are left with
sec^2x-1
*You can use the formula 1+tan^2(theta)=sec^2(theta)
tan^2(theta)=sec^2(theta)-1
=tan^2(theta)