8-4 Relationships between the Functions
 
 
 All of the following relationships need to be memorized.  These are the most commonly used functions that appear in many upper level calculus courses.  Please make every attempt to really know them!!
 
 
Reciprocal Relationships
csc q = 1/sin q sec q = 1/cos q cot q = 1/tan q
 
 
Negative Relationships
sin(-q) = -sin q cos(-q) = cos q
csc (-q) = -csc q sec(- q) = sec q
tan(-q ) = -tan q cot(- q) = -cot q
 
 
Pythagorean Relationships
sin2 q  + cos2 q  = 1 1 + tan2 q  = sec2 q 1 + cot2 q  = csc2 q
 
 
 
Cofunction Relationships
sin q = cos(90o - q ) cos q = sin(90o - q)
tan q = cot(90o - q) cot q = tan(90o - q)
sec q = csc(90o - q) csc q = sec(90o - q)

Any relationship that is true for all values of the variable for which each side is defined is called an identity.  Each of the relationships above represent a trigonometric identity.  We can use trig identities to simplify trig expressions, prove other identities and solve more complex trig equations.
 
 
Simplifying trig expressions
 
1) simplify: sin y cot y 
= sin y (cos y/sin y)  (Basic identity)
= cos y (Basic algebra)
 
2)  simplify: csc2 y(1 - cos2 y)
= csc2 y (sin2 y)  (pythagorean)
= csc2 y(1/csc2 y) (reciprocal)
= 1  (basic algebra)
 
3)  simplify:  tan x(cot x + tan x)
= 1 + tan2 x  (distribute) (tan and cot are reciprocals)
= sec2 x   (pythagorean)
 
4)  simplify:  cos x cot x + sin x
 
5)  simplify: 
 
6)  simplify: 
 

 Proving Identities
 You can use trig identities to prove other statements are identities.  You may simplify either the left side of the equation or the right side of the equal sign.  Your goal is to make it look like the other side of the equation.  You may even work on both sides.  It is probably a good idea to simplify the more complicated side.  Also, remember your basic algebra skills, like getting a common denominator and factoring.  Keep in mind where you are headed.  If cosine is on the right side, then when working on the left side, try to write it in terms of cosine.  Use the identities.  They will help!  If all else fails, try writing everything in terms of the sine and cosine.  Good luck!!
 
1)  Prove:  tan x + cot x = tan x csc2 x
                            Solution:
 
 
2)  Prove: 
Solution:
 
 
3)  Prove: 
 
Solution:
 
 
4)  Prove: 
Solution:
 
 
5) Prove: 
Solution:

 
 That should give you some techniques in working with identities.  Now, let's head onto more difficult trig equations.